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Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

39278960
72621
10.1038/s41598-024-72621-7
Article
On degree-based operators and topological descriptors of molecular graphs and their applications to QSPR analysis of carbon derivatives
Khan Abdul Rauf 1
Bhatti Saad Amin 1
Tawfiq Ferdous 2
Siddiqui Muhammad Kamran 3
Hussain Shahid 4
Ali Mustafa Ahmed mustafa@snu.edu.so

5
1 https://ror.org/023a7t361 grid.448869.f 0000 0004 6362 6107 Department of Mathematics, Faculty of Sciences, Ghazi University, Dera Ghazi Khan, 32200 Pakistan
2 https://ror.org/02f81g417 grid.56302.32 0000 0004 1773 5396 Mathematics Department, College of Science, King Saud University, P.O. Box 22452, Riyadh, 11495 Saudi Arabia
3 https://ror.org/00nqqvk19 grid.418920.6 0000 0004 0607 0704 Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore, Pakistan
4 https://ror.org/016st3p78 grid.6926.b 0000 0001 1014 8699 Energy Engineering Division, Department of Engineering Science and Mathematics, Lulea University of Technology, Lulea, Sweden
5 https://ror.org/03f3jde70 grid.412667.0 0000 0001 2156 6060 Department of Mathematics, Faculty of Science, Somali National University, Mogadishu, Somalia
15 9 2024
15 9 2024
2024
14 215434 5 2024
9 9 2024
© The Author(s) 2024
2024
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This work initiates a concept of reduced reverse degree based RRDM-Polynomial for a graph, and differential and integral operators by using this RRDM-Polynomial. In this study twelve reduced reverse degree-based topological descriptors are formulated using the RRDM-Polynomial. The topological descriptors, denoted as TD’s, are numerical invariants that offer significant insights into the molecular topology of a molecular graph. These descriptors are essential for conducting QSPR investigations and accurately estimating physicochemical attributes. The structural and algebraic characteristics of the graphene and graphdiyne are studied to apply this methodology. The study involves the analysis and estimation of Reduced reverse degree-based topological descriptors and physicochemical features of graphene derivatives using best-fit quadratic regression models. This work opens up new directions for scientists and researchers to pursue, taking them into new fields of study.

Keywords

Molecular graph
Combinatorics
RRDM-Polynomial
Integral operator
Reduced reverse degree-based topological descriptors
α-Graphyne
β-Graphyne
α-Graphdiyne
QSPR analysis
Physicochemical properties
Subject terms

Applied mathematics
Physical chemistry
issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

The discovery of graphene has revolutionized materials science, igniting curiosity in additional 2D carbon allotropes like graphynes and grapevines. In 1987, these materials were initially predicted1. The significance of two-dimensional graphene, graphyne, and graphdiyne derivatives of graphite structures is growing due to its encouraging characteristics, which include energy level alignment, charge carrier mobilities, and tunable band gaps2,3. Two-dimensional carbon allotrope families with acetylenic groups joining benzenoid-like hexagonal rings are referred to as graphynes or graphdiynes4. The useful approach for forecasting drug-drug interactions based on knowledge graph neural networks and molecular substructures was presented by Chen et al.5.

Graphene is a two-dimensional in-nature sheet composed of hexagons and sp2 hybridized carbon atoms6. Because carbon atoms are so adaptable, it is theoretically possible to create carbon allotropes by modifying the periodic patterns within networks of sp3-, sp2-, and sp-hybridized carbon atoms7,8. The main distinction between graphynes and grapevines is the presence of one or two acetylenic groups. Increasing the amount of acetylenic groups can theoretically lead to an almost endless number of related structures. Graphene has attracted a lot of attention because of its remarkable mechanical, chemical, thermal, electrical, and physical properties9,10.

Recently, graphene-related research has received a lot of attention because it was honoured with the 2010 Nobel Prize in Physics for its “groundbreaking relating to the two-dimensional (2D) substance”11. When the bonds between the three coordinated atoms in a graphene layer are swapped with carbyne chains, the states of the sp2 atoms stay equivalent and graphene layers are formed12.

Graphene can be converted into two-dimensional materials called graphynes by adding acetylenic connections to a honeycomb structure that contains sp hybridized C-atoms. These structures possess an array of electrical, optical, and mechanical capabilities due to the presence of acetylenic groups within them13. Three-dimensional reconstruction and geometric morphology analysis of lunar small craters within the Yutu-2 rover’s patrol range were examined by Xu et al.14.

Graphdiyne is a novel synthetic carbon-based nanomaterial with sp and sp2 hybridized carbon atoms derived from acetylenic groups and benzene rings. This new material has been created by synthesizing sp2 and sp-hybridized carbon atoms15. The α, β, and γ-type structures are the most notable configurations of graphyne and its derivatives16,17. Among them, the α-type structures’ topological descriptors have been examined18,19. Because these structures have such vital uses, a research study of the topological descriptors of these different networks would be crucial to compare and contrast the complexities of these structures20. The specular removal of industrial metal Objects without changing lighting configuration was analysed by Chen et al.21.

Graphene derivatives, such as graphyne and graphdiyne, have received a lot of attention in mathematics and science because of their outstanding optical, electrical, and mechanical capabilities. These materials have various mathematical applications, including Graphyne and graphdiyne are combinatorial graphs in combinatorics that possess features that can be examined through graph theory. Xu et al.22 used attentive GAN to analyse and remove highlights from a single greyscale image. Graph-theoretic concepts including graph colouring, graph embedding, and graph isomorphism are studied about these materials. Graphyne and graphdiyne are used to analyze the behaviour of different topological descriptors since they are numerical invariants that provide information on the topology of a molecular graph.

The graphene derivatives are helpful in quantum mechanics to examine how electrons behave in a two-dimensional lattice structure. Quantum mechanics can be applied to investigate the electrical characteristics of these materials. Tight-binding approximation (TBA) and density functional theory (DFT) are two methodologies for predicting these materials’ electronic structures23.

These materials are significant for further research and development because graphene derivatives, such as graphyne and graphdiyne, have a wide range of mathematical applications across several fields. Physical attributes of both structures are determined by their topological descriptors24,25. It has been found that adding metallic elements and metallic binding to these carbonaceous materials significantly increases the compressive energy of the metallic element. These substances can bind molecules with substantially reduced sorption energies because of their almost uniform metallic atom charges26. The estimated maximum sorption enthalpy for near-room temperature element storage (3.6 kcal/mol) roughly matches the crucial computed enthalpies for element sorption, which vary from 3.5 to 2.8 kcal/mol. Planar carbon allotropes with tunable bandgaps made possible by changing the quantity of alphabetically connected bridging units show promise for use in microelectronics. The cycle-consistent generative adversarial network-based nighttime road scene image enhancement was discussed by Jia et al.27.

A molecular graph is a graphic representation of a chemical compound’s structural formula in which the vertices, or nodes, stand in for atoms and the edges indicate bonds between atoms28,29. If G is a graph, then the fundamental symbols and definitions that are utilized, like dλ, which stands for the degree of the vertex λ, are taken from the book that is mentioned in30. Topological descriptors (TD) are used to determine the graphical structures of chemical compounds, and graph invariants could be used. TD are fundamentally represented by converting a chemical graph to a numerical value. Wiener suggests to use TD in 1947. He originally reported this index (W) on trees and examined how it was used to correlate the physical features of alcohols, alkanes, and related complexes31,32.

Topological descriptors, which are structural invariants grounded in molecular graphs and which capture the fundamental connectivity of molecular networks, have drawn a lot of interest lately due to their applications in the fields of quantitative structure-activity and quantitative structure-property relations (QSPR) relations33,34.

The predictive potential of distance-based and spectrum-based topological descriptors for measuring the π electron energy of benzenoid hydrocarbons was discussed by Hayat et al.35–38 and Malik et al.39, with applications to carbon nanotubes and boron α- and triangular-nanotubes, respectively. Cheminformatics is an emerging discipline that supports QSPRs, which are frequently employed to forecast the bioactivities and characteristics of chemical compounds40–43.

In QSPR research, topological descriptors combined with entropy measures may be a more effective tool. Quality tests of spectrum-based valency and distance-based molecular descriptors for polycyclic aromatic and benzenoid hydrocarbons with applications to carbon nanotubes and nanocones were conducted by Hayat et al.44–47.

Physicochemical and topological descriptors have been utilized to predict the bioactivity of organic compounds48–50. In a chemical graph, atoms or compounds are represented by the vertices, and their chemical interactions are represented by the contacts. Recently, Eryaşar et al.51 introduced new formulas and new bounds for the First and Second Zagreb descriptors of Phenylenes. Öztürk Sözen et al.53,54 investigated an algebraic approach to calculate some topological descriptors and QSPR analysis of some novel drugs used in the treatment of breast cancer and COVID-19. Twisted relative rota-Baxter operators on Leibniz conformal algebras were studied by Guo et al.52.

Topological descriptors employing Reduced Reverse Degree M-polynomial have not been investigated for graphenes’ α and β structures.

TD delineates the graph’s structure, while numerical graph invariants. According to West et al.55, the degree in any vertex is represented with dλ or d(λ) and represents the number of edges that intersect that vertex λ. Many researchers are currently performing QSPR investigations of various molecules because it is a more economical way to test compounds than evaluating them56,57.

In this article, we have provided results for the computation of reduced reverse degree-based topological descriptors (TD) for graphyne and graphdiyne. Notably, our work introduced a novel approach for analyzing TD which is Reduced Reverse Degree M-polynomial builds upon the foundation laid by Zaman et.al58.

Objectives of the study

This study’s primary goal is to provide the reduced reverse degree-based graph polynomial and integral and differential operators. Using these operators, the formulation of topological descriptors can be determined. Our goal is to calculate the physical and chemical properties of certain Molecular Graphs using this methodology.

Novelity in the study

We present a novel notion called the reduced reverse degree-based graph polynomial. Polynomial differential and integral operators are formulated based on this. This enables us to create topological descriptors depending on the reduced reverse degree. To execute our methodology, we obtained molecular graphs of α-Graphyne, β-Graphyne, and α-Graphdiyne, as depicted in Figs. 1, 2 and 3. We then assessed the physicochemical attributes of these Molecular Graphs using the data computed by our methodology.

Material and methodolgy

Structure of graphyne and graphdiyne

Graphene’s related graphyne and graphdiyne are carbon-based materials with different carbon atom arrangements giving them different structures and physical characteristics. Graphene and graphyne are two-dimensional lattices made of carbon atoms, but graphyne has extra triple bonds connecting some of the carbon atoms. A variation of graphyne known as graphdiyne has two successive triple bonds between some carbon atoms in its structure. Like graphene, both graphyne and graphdiyne are carbon derivatives, but they differ in their bonding arrangements, giving them distinct characteristics. Graphene and graphdiyne belong to the family of carbon-based materials that provide different bonding patterns to increase the versatility of graphene and give it unique qualities.

One carbon allotrope is graphyne. Its structure consists of a planar sheet of sp and sp2 linked carbon atoms organized in a crystal lattice, one atom thick. Graphyne is a graphene derivative where the hexagons are connected by acetylenic bonds, as seen in Figs. 1 and 3. In 1987, Baughman et al.59 made the initial proposal for graphyne as part of a larger study into the characteristics of novel forms of carbon that had been reported occasionally but not thoroughly examined. Graphyne’s unique electrical structure distinguishes it from other carbon-based materials like diamond and graphite. The most common form of graphyne is α-Graphyne and β-Graphyne.

Significant scientific effort has been directed towards other two-dimensional materials following the discovery of graphene and the prediction of graphyne. Graphdiyne is one among those, it is a variation of graphyne that has two acetylenic links in each unit cell instead of graphyne’s single bond, as shown in Fig. 2. The acetylenic links double the length of the carbon chains that connect the hexagonal rings. In 1997, Haley et al.60 made the first prediction about graphdiyne. Initially, the material was created by synthesizing it from related organic molecules and estimating its qualities using computational simulations of associated materials. Graphdiyne belongs to the graphyne family, but because of its distinctive features, it is usually treated as an independent entity61,62.

Computational techniques

A connected graph G with vertex and edge sets V(G) and E(G), respectively, can be used to simulate a chemical structure63. The number of edges of G incident with vertex λ is called the degree of vertex λ. The idea of reverse degree vertex RD(λ), introduced by Kulli64, is defined as follows:RD(λ)=δ(G)-d(λ)+1

Inspiring by this concept Ravi65 defines the reduced reverse degree as:RRD(λ)=δ(G)-d(λ)+2

The degree and reduced reverse degree of an atom is denoted by d(λ) and RRD(λ) of the vertex λ∈V(G) respectively, whereas the maximum degree and maximum reduced reverse degree over all the vertices of G is denoted by δ(G) and χ(G) respectively. Consider the set RRD={(i,j∈NXN):1≤i≤j≤χ}. We denote RRD(i,j)={λτ∈E(G):RRD(λ)=i,RRD(τ)=j}. The modified reverse degree66 is defined asMk(RD)=δ(G)-d(λ)+k:k≤d(λ)δ(G)-d(λ)+k(mod(δ(G)):k>d(λ)

The M-Polynomial67 is defined as follows:1 M(G;k,q)=∑i≤jρ(i,j)kiqj

where ρ(i,j) denotes the number of edges in the graph G for any pair of indices i and j where i≤j.

In this study, we have initiated the Reduced reverse degree based RRDM-Polynomial, defined as follows in consensus with the previous studies of M-Polynomial68.2 RRDM(G;l,m)=∑i≤jν(i,j)limj

ν(i,j) is counted as number of edges λτ∈E(G) such that {RRD(λ),RRD(τ)}={i,j}.

Table 1 contains the formulas that connect the reduced reverse degree-based topological descriptors to the RRDM-Polynomial. In Table 1, operators are defined as below:3 Dl=l∂(RRDM(G;l,m))∂l

4 Dm=m∂(RRDM(G;l,m))∂m

5 Il=∫0l1z(RRDM(G;z,m))dz

6 Im=∫0m1z(M(G;l,z))dz

7 J(g(l,m))=g(l,l)

8 Qα=lαg(l,m

Table 1 Topological descriptors derivation formula from RRDM-polynomial.

Topological index	Derived formula	
RRDM1	(Dl+Dm)(RRDM(G))|l=m=1	
RRDM2	(DlDm)(RRDM(G)|l=m=1	
RRDF	(Dl2+Dm2)(RRDM(G)|l=m=1	
RRDHM1	(Dl+Dm)2(RRDM(G)|l=m=1	
RRDHM2	(DlDm)2(RRDM(G)|l=m=1	
RRDσ	(Dl-Dm)2(RRDM(G))|l=m=1	
RRDmM2	(IlIm)(RRDM(G)|l=m=1	
RRDReZG3	(DlDm)(Dl+Dm)(RRDM(G)|l=m=1	
RRDSDD	(DlIm+IlDm)(RRDM(G)|l=m=1	
RRDH	2JIl(RRDM(G)|l=m=1	
RRDI	IlJDlDm(RRDM(G)|l=m=1	
RRDA	Il3Q-2JDl3Dm3(RRDM(G)|l=m=1	

Results

This section contains the findings of the study. This section begins with the formulation of analytical formulas for a variety of reduced reverse degree-based topological descriptors of α and β types for graphene, graphyne, and graphdiyne nanoribbons. Next, using the derived topological descriptors, probabilistic numerical values for these nanoribbons are computed and tabulated. The resulting numerical values are then graphically shown using QSPR modeling of various structures with various topological descriptors and physiochemical features. Figures 1, 2 and 3 displays the structures of the different graphene derivatives that we are interested in. We shall use the notaions GYα and GYβ to denote α-graphyne and β graphyne, GDα to denote α-graphdiyne structures.

Reduce reverse degree based RRDM-polynomials for α-graphyne

Using Fig. 1, the reduced reverse degree-based edge division of α-graphyne is described below in Table 2.Figure 1 Structure of α-graphyne.

Table 2 Reduce reverse degree based edge partition of α-graphyne.

RRDE(λ,τ)	Cardinality	
E(3,3)	6rt+3r+9t	
E(3,2)	12rt-6r-6t	

The Reverse Degree-BasedM-Polynomial forGYα(r,t)

Let GYα(r,t) be graph of α-graphyne. Let RRD(i,j) be the set of all edges with a reduced reverse degree of end vertices i, j and νi,j be the number of edges in RRD(i,j).RRD(i,j)={λτ∈E(G):RRD(λ)=i,RRD(τ)=j}RRD(3,3)={λτ∈E(G):RRD(λ)=3,RRD(τ)=3}RRD(3,2)={λτ∈E(G):RRD(λ)=3,RRD(τ)=2}

From Fig. 1 and Table 2, it is clear thatν(3,3)=6rt+3r+9t,ν(3,2)=12rt-6r-6t

The RRDM-Polynomial of GYα(r,t) is obtained as followRRDM(GYα)=∑i≤jν(i,j)limj=ν(3,3)l3m3+ν(3,2)l3m2

Putting values of ν(i,j), we obtain9 RRDM(GYα;l,m)=(6rt+3r+9t)l3m3+(12rt-6r-6t)l3m2

The differential operators forGYα(r,t)

In view of Table 1, using operators (3) and (4) along with Eq. (9), we get10 Dl(RRDM(GYα;l,m))=(18rt+9r+27t)l3m3+(36rt-18r-18t)l3m2

11 Dm(RRDM(GYα;l,m))=(18rt+9r+27t)l3m3+(24rt-12r-12t)l3m2

The integral operators forGYα(r,t)

In view of Table 1, using operators (5) and (6) along with Eq. (9), we get12 Il(RRDM(GYα;l,m))=(2rt+r+3t)l3m3+(4rt-2r-2t)l3m2

13 Im(RRDM(GYα;l,m))=(2rt+r+3t)l3m3+(6rt-3r-3t)l3m2

Topological descriptors ofα-Graphyne usingRRDM-Polynomial approach

Reduced Reverse 1st Zagreb Index ofGYα(r,t)

In view of Table 1, adding Eqs. (10) and (11), we get(Dl+Dm)(RRDM(GYα;l,m))=(18rt+9r+27t)l3m3+(36rt-18r-18t)l3m2+(18rt+9r+27t)l3m3+(24rt-12r-12t)l3m2(Dl+Dm)(RRDM(GYα;l,m))|l=m=1=[(36rt+18r+54t)l3m3+(60rt-30r-30t)l3m2]|l=m=1RRDM1(GYα)=96rt-12r+24t

Reduced Reverse 2nd Zagreb Index ofGYα(r,t)

In view of Table 1, applying differential operator on Eq. (11), we haveDl[(Dm(RRDM(GYα;l,m))]=Dl[(18rt+9r+27t)l3m3+(24rt-12r-12t)l3m2](DlDm)(RRDM(GYα;l,m))|l=m=1=[(54rt+27r+81t)l3m3+(72rt-36r-36t)l3m2]|l=m=1RRDM2(GYα)=126rt-9r+45t

Reduced Reverse Forgotten Index ofGYα(r,t)

In view of Table 1, applying differential operators on Eqs. (10) and (11), we haveDl2(RRDM(GYα;l,m))=(54rt+27r+81t)l3m3+(108rt-54r-54t)l3m2Dm2(RRDM(GYα;l,m))=(54rt+27r+81t)l3m3+(48rt-24r-24t)l3m2(Dl2+Dm2)(M(GGYα;x,y))=(54rt+27r+81t)l3m3+(108rt-54r-54t)l3m2+(54rt+27r+81t)l3m3+(48rt-24r-24t)l3m2(Dl2+Dm2)(RRDM(GYα;l,m))=(108rt+54r+162t)l3m3+(156rt-78r-78t)l3m2(Dl2+Dm2)(RRDM(GYα;l,m))|l=m=1=(108rt+54r+162t)l3m3+(156rt-78r-78t)l3m2|l=m=1

After putting limits, we haveRRDF(GYα)=264rt-24r+84t

Reduced Reverse Hyper 1st Zagreb Index ofGYα(r,t)

In the view of Table 1, and after applying differential operator on Eqs. (10) and (11), we have(Dl2+Dm2)(RRDM(GYα;l,m))=(54rt+27r+81t)l3m3+(108rt-54r-54t)l3m2+(54rt+27r+81t)l3m3+(48rt-24r-24t)l3m2(2DlDm)(RRDM(GYα;l,m))=(108rt+54r+162t)l3m3+(144rt-72r-72t)l3m2(Dl+Dm)2(GYα;l,m))|l=m=1=(216rt+108r+324t)l3m3+(300rt-150r-150t)l3m2|l=m=1

After putting limits, we haveRRDHM1(GYα)=516rt-42r+174t

Reduced Hyper 2nd Zagreb Index ofGYα(r,t)

In the view of Table 1 after applying differential operator on Eq. (11), we have(Dl2Dm2)(RRDM(GYα;l,m))=Dl2[(54rt+27r+81t)l3m3+(48rt-24r-24t)l3m2](Dl2Dm2)(RRDM(GYα;l,m))|l=m=1=(486rt+243r+729t)l3m3+(432rt-216r-216t)l3m2|l=m=1

After putting limits, we haveRRDHM2(GYα)=918rt+27r+513t

Reduced Reverse Sigma Index ofGYα(r,t)

In the view Table 1, and after applying differential operator on Eqs. (10) and (11), we have(Dl2+Dm2)(RRDM(GYα;l,m))=54rt+27r+81t)l3m3+(108rt-54r-54t)l3m2+(54rt+27r+81t)l3m3+(48rt-24r-24t)l3m2(2DlDm)(RRDM(GYα;l,m))=(108rt+54r+162t)l3m3+(144rt-72r-72t)l3m2(Dl-Dm)2(RRDM(GYα;l,m))|l=m=1=(12rt-6r-6t)l3m2|l=m=1

After putting limits, we haveRRDσ(GYα)=12rt-6r-6t

Reduced Reverse Second Modified Zagreb Index ofGYα(r,t)

From Table 1, Second Modified Zagreb Index for RRDM-Polynomial is(IlIm)(RRDM(GYα;l,m))=Il[Im(RRDM(GYα;l,m)]=Il[(2rt+r+3t)l3m3+(6rt-3r-3t)l3m2]

After operating integral operator Il (5), we have(IlIm)(RRDM(GYα;l,m))|l=m=1=23rt+13r+tl3m3+(2rt-r-t)l3m2|l=m=1

After putting limits, we haveRRDmM2(GYα)=83rt-23r

Reduced Reverse Redefined Third Zagreb Index

In view of Table 1, Reduced Reverse Redefined Third Zagreb Index for RRDM-Polynomial is computed as follow(Dl+Dm)(M(GYα;l,m))=(36rt+18r+54t)l3m3+(60rt-30r-30t)l3m2Dm(Dl+Dm)(RRDM(GYα;l,m)=(108rt+54r+162t)l3m3+(120rt-60r-60t)l3m2(DlDm)(Dl+Dm)(RRDM(GYα;l,m))=(324rt+162r+486t)l3m3+(360rt-180r-180t)l3m2(DlDm)(Dl+Dm)(RRDM(GYα;l,m))|l=m=1=(324rt+162r+486t)l3m3+(360rt-180r-180t)l3m2)|l=m=1

After putting limits, we haveRRDReZG3(GYα;r,t)=684rt-18r+306t

Reduced Reverse Symmetric Division Degree Index ofGYα(r,t)

In view Table 1 using Eq. (13), we have(Im)(RRDM(GYα;l,m))=(2rt+r+3t)l3m3+(6rt-3r-3t)l3m2

After applying differential operator Dl (3), we have(DlIm)(RRDM(GYα;l,m))=(6rt+3r+9t)l3m3+(18rt-9r-9t)l3m2

Also from Eq. (11)Dm(RRDM(GYα;l,m))=(18rt+9r+27t)l3m3+(24rt-12r-12t)l3m2

After operating integral operator Il (5), we haveIlDm(RRDM(GYα;l,m))=(6rt+3r+9t)l3m3+(8rt-4r-4t)l3m2(DlIm+IlDm)(RRDM(GYα;l,m))=(12rt+6r+18t)l3m3+(26rt-13r-13t)l3m2(DlIm+IlDm)(RRDM(GYα;l,m))|l=m=1=(12rt+6r+18t)l3m3+(26rt-13r-13t)l3m2|l=m=1RRDSDD(GYα)=48rt-7r+5t

Reduced Reverse Harmonic Index ofGYα(r,t)

In view of Table 1, applying operator (8) on Eq. (9), we haveJRRDM(GYα;l,m)=(6rt+3r+9t)l6+(12rt-6r-6t)l5

After operating integral operator Il (5), we haveIlJRRDM(GYα;l,m)=rt+12r+32tl6+125rt-65r-65tl52IlJRRDM(GYα;l,m)=(2rt+r+3t)l6+245rt-125r-125tl52IlJRRDM(GYα;l,m)|l=m=1=(2rt+r+3t)l6+245rt-125r-125tl5|l=1

RRDH(GYα)=345rt-75r+35t

Reduced Reverse Inverse Sum Index ofGYα(r,t)

From Table 1, Harmonic Index for RRDM-Polynomial isDlDm(RRDM(GYα;l,m))=(54rt+27r+81t)l3m3+(72rt-36r-36t)l3m2JDlDm(RRDM(GYα;l,m))=(54rt+27r+81t)l6+(72rt-36r-36t)l5

After operating integral operator Il (5), we haveIlJDlDm(RRDM(GYα;l,m))=9rt+92r+272tl6+725rt-365r-365tl5IlJDlDm(RRDM(GYα;l,m))|l=1=9rt+92r+272tl6+725rt-365r-365tl5|l=1

RRDI(GYα)=1175rt-2710r+6310t

Reduced Reverse Augmented Index ofGYα(r,t)

In view Table 1 applying operator Dm3 on Eq. (9), we haveDm2(RRDM(GYα;l,m))=(54rt+27r+81t)l3m3+(48rt-24r-24t)l3m2Dm3(RRDM(GYα;l,m))=(162rt+81r+243t)l3m3+(96rt-48r-48t)l3m2

After operating differential operator Dl3, we haveDl3Dm3(RRDM(GYα;l,m))=(4374rt+2187r+6561t)l3m3+(2592rt-1296r-1296t)l3m2JDl3Dm3(RRDM(GYα;l,m))=(4374rt+2187r+6561t)l6+(2592rt-1296r-1296t)l5Q-2JDl3Dm3(RRDM(GYα;l,m))=l-2[(4374rt+2187r+6561t)l6+(2592rt-1296r-1296t)l5]Q-2JDl3Dm3(RRDM(GYα;l,m))=(4374rt+2187r+6561t)l4+(2592rt-1296r-1296t)l3

After operating Integral operator Il3, we haveIl3Q-2JDl3Dm3(RRDM(GYα;l,m))=218732rt+218764r+656164tl4+(96rt-48r-48t)l3Il3Q-2JDl3Dm3(RRDM(GYα;l,m))|l=1=[218732rt+218764r+656164t)l4+(96rt-48r-48t)l3]|l=1RRDA(GYα)=525932rt-88564r+348964t

Reduced reverse degree based RRDM-polynomials for α-graphdiyne

Using Fig. 2, the reduced reverse degree-based edge division of α-graphdiyne is described below in Table 3.Figure 2 Structure of α-graphdiyne.

Table 3 Reduced reverse degree based edge partition of α-graphdiyne.

RRDE(λ,τ)	Cardinality	
E(3,3)	18rt+r+11t	
E(3,2)	12rt-6r-6t	

Reduce Reverse Degree-BasedM-Polynomial forGDα(r,t)

Let GDα(r,t) be graph of α-graphdiyne . Let RRD(i,j) be the set of all edges with a reduced reverse degree of end vertices i, j and νi,j be the number of edges in RRD(i,j).RRD(i,j)={λτ∈E(G):RRD(λ)=i,RRD(τ)=j}RRD(3,3)={λτ∈E(G):RRD(λ)=3,RRD(τ)=3}RRD(3,2)={λτ∈E(G):RRD(λ)=3,RRD(τ)=2}

From Fig. 2 and Table 3, it is clear that ν(3,3) = 18rt+r+11t, ν(3,2) = 12rt-6r-6t

The RRDM-Polynomial of GDα(r,t) is obtained as followRRDM(GDα)=∑i≤jν(i,j)limj=ν(3,3)l3m3+ν(3,2)l3m2

Putting values of ν(i,j), we obtain14 RRDM(GDα;l,m)=(18rt+r+11t)l3m3+(12rt-6r-6t)l3m2

The differential operators forGDα(r,t)

In view of Table 1, using operators (3) and (4) along with Eq. (14), we get15 Dl(RRDM(GDα;l,m))=(54rt+3r+33t)l3m3+(36rt-18r-18t)l3m2

and16 Dm(RRDM(GDα;l,m))=(54rt+3r+33t)l3m3+(24rt-12r-12t)l3m2

The integral operators forGDα(r,t)

In view of Table 1, using operators (5) and (6) along with Eq. (14), we get17 Il(RRDM(GDα;l,m))=6rt+13r+113tl3m3+(4rt-3r-3t)l3m2

18 Im(RRDM(GDα;l,m))=6rt+13r+113tl3m3+(6rt-3r-3t)l3m2

Topological descriptors ofα-graphdiyne usingRRDM-Polynomial approach

Reduced Reverse 1st Zagreb Index ofGDα(r,t)

In view of Table 1, adding Eqs. (15) and (16), we get(Dl+Dm)(RRDM(GDα;l,m))=(54rt+3r+33t)l3m3+(36rt-18r-18t)l3m2+(54rt+3r+33t)l3m3+(24rt-12r-12t)l3m2(Dl+Dm)(RRDM(GDα;l,m))|l=m=1=[(108rt+6r+66t)l3m3+(60rt-30r-30t)l3m2]|l=m=1RRDM1(GDα)=168rt-24r+36t

Reduced Reverse 2nd Zagreb Index ofGDα(r,t)

In view of Table 1, applying differential operator on Eq. (16), we haveDl[(Dm(RRDM(GDα;l,m))]=Dl[(54rt+3r+33t)l3m3+(24rt-12r-12t)l3m2](DlDm)(RRDM(GDα;l,m))|l=m=1=[(162rt+9r+99t)l3m3+(72rt-36r-36t)l3m2]|l=m=1RRDM2(GDα)=234rt-27r+63t

Reduced Reverse Forgotten Index ofGDα(r,t)

In view of Table 1, applying differential operators on Eqs. (15) and (16), we haveDl2(RRDM(GDα;l,m))=(162rt+9r+99t)l3m3+(108rt-54r-54t)l3m2Dm2(RRDM(GDα;l,m))=(162rt+9r+99t)l3m3+(48rt-24r-24t)l3m2(Dl2+Dm2)(M(GGDα;x,y))=(162rt+9r+99t)l3m3+(108rt-54r-54t)l3m2+(162rt+9r+99t)l3m3+(48rt-24r-24t)l3m2(Dl2+Dm2)(RRDM(GDα;l,m))=(324rt+18r+198t)l3m3+(156rt-78r-78t)l3m2(Dl2+Dm2)(RRDM(GDα;l,m))|l=m=1=(324rt+18r+198t)l3m3+(156rt-78r-78t)l3m2|l=m=1

After putting limits, we haveRRDF(GDα)=480rt-60r+120t

Reduced Reverse Hyper 1st Zagreb Index ofGDα(r,t)

In the view of Table 1, and after applying differential operator on Eqs. (15) and (16), we have(Dl2+Dm2)(RRDM(GDα;l,m))=(162rt+9r+99t)l3m3+(108rt-54r-54t)l3m2+(162rt+9r+99t)l3m3+(48rt-24r-24t)l3m2(2DlDm)(RRDM(GDα;l,m))=(324rt+18r+198t)l3m3+(144rt-72r-72t)l3m2(Dl+Dm)2(RRDM(GDα;l,m))|l=m=1=(648rt+36r+396t)l3m3+(144rt-72r-72t)l3m2|l=m=1

After putting limits, we haveRRDHM1(GDα)=948rt-114r+246t

Reduced Reverse Hyper 2nd Zagreb Index ofGDα(r,t)

In the view of Table 1, and after applying differential operator Dl2 on Eq. (16), we have(Dl2Dm2)(RRDM(GDα;l,m))=Dl2[(162rt+9r+99t)l3m3+(48rt-24r-24t)l3m2](Dl2Dm2)(RRDM(GDα;l,m))|l=m=1=(1458rt+81r+891t)l3m3+(432rt-216r-216t)l3m2|l=m=1

After putting limits, we haveRRDHM2(GDα)=1890rt-135r+675t

Reduced Reverse Sigma Index ofGDα(r,t)

In the view Table 1, and after applying differential operator on Eqs. (15) and (16), we have(Dl2+Dm2)(RRDM(GDα;l,m))=(162rt+9r+99t)l3m3+(108rt-54r-54t)l3m2+(162rt+9r+99t)l3m3+(48rt-24r-24t)l3m2(2DlDm)(RRDM(GDα;l,m))=(324rt+18r+198tl3m3+(144rt-72r-72t)l3m2(Dl-Dm)2(RRDM(GDα;l,m))|l=m=1=(12rt-6r-6t)l3m2|l=m=1

After putting limits, we haveRRDσ(GDα)=12rt-6r-6t

Reduced Reverse Second Modified Zagreb Index ofGDα(r,t)

In view Table 1, using Eq. (18)(IlIm)(RRDM(GDα;l,m))=Il[Im(RRDM(GDα;l,m)]=Il6rt+13r+113tl3m3+(6rt-3r-3t)l3m2

After operating Integral operator Il (5), we have(IlIm)(RRDM(GDα;l,m))|l=m=1=2rt+19r+119tl3m3+(2rt-r-t)l3m2|l=m=1

After putting limits, we haveRRDmM2(GDα)=4rt-89r+29t

Reduced Reverse Redefined Third Zagreb Index ofGDα(r,t)

In view of Table 1, Reduced Reverse Redefined Third Zagreb Index for RRDM-Polynomial is computed as follow(Dl+Dm)(M(GDα;l,m))=(108rt+6r+66t)l3m3+(60rt-30r-30t)l3m2Dm(Dl+Dm)(RRDM(GDα;l,m)=(324rt+18r+198t)l3m3+(120rt-60r-60t)l3m2(DlDm)(Dl+Dm)(RRDM(GDα;l,m))=(972rt+54r+594t)l3m3+(360rt-180r-180t)l3m2(DlDm)(Dl+Dm)(RRDM(GDα;l,m))|l=m=1=(972rt+54r+594t)l3m3+(360rt-180r-180t)l3m2)|l=m=1

After putting limits, we haveRRDReZG3(GDα)=1332rt-126r+414t

Reduced Reverse Symmetric Division Degree Index ofGDα(r,t)

In view Table 1 using Eq. (18), we have(Im)(RRDM(GDα;l,m))=6rt+13r+113tl3m3+(6rt-3r-3t)l3m2

After applying operator Dl (3), we have(DlIm)(RRDM(GDα;l,m))=(18rt+r+11t)l3m3+(18rt-9r-9t)l3m2

Also from Eq. (16)Dm(RRDM(GDα;l,m))=(54rt+3r+33t)l3m3+(24rt-12r-12t)l3m2

After operating integral operator Il (5), we haveIlDm(RRDM(GDα;l,m))=(18rt+r+11t)l3m3+(8rt-4r-4t)l3m2(DlIm+IlDm)(RRDM(GDα;l,m))=(36rt+2r+22t)l3m3+(26rt-13r-13t)l3m2(DlIm+IlDm)(RRDM(GDα;l,m))|l=m=1=(36rt+2r+22t)l3m3+(26rt-13r-13t)l3m2|l=m=1RRDSDD(GDα)=62rt-11r+9t

Reduced Reverse Harmonic Index ofGDα(r,t)

In view of Table 1, applying operator (8) on Eq. (14), we haveJRRDM(GDα;l,m)=(18rt+r+11t)l6+(12rt-6r-6t)l5

After operating integral operator Il (5), we haveIlJRRDM(GDα;l,m)=3rt+16r+116tl6+125rt-65r-65tl52IlJRRDM(GDα;l,m)=6rt+13r+113tl6+245rt-125r-125tl52IlJRRDM(GDα;l,m)|l=m=1=6rt+13r+113tl6+245rt-125r-125tl5|l=1RRDH(GDα)=545rt-3115r+195t

Reduced Reverse Inverse Sum Index ofGDα(r,t)

From Table 1, Harmonic Index for RRDM-Polynomial isDlDm(RRDM(GDα;l,m))=(162rt+9r+99t)l3m3+(72rt-36r-36t)l3m2JDlDm(RRDM(GDα;l,m))=(162rt+9r+99t)l6+(72rt-36r-36t)l5

After operating integral operator Il (5), we haveIlJDlDm(RRDM(GDα;l,m))=27rt+32r+332tl6+725rt-365r-365tl5IlJDlDm(RRDM(GDα;l,m))|l=1=27rt+32r+332tl6+725rt-365r-365tl5|l=1RRDI(GDα)=2075rt-5710r+9310t

Reduced Reverse Augmented Index ofGDα(r,t)

In view Table 1 applying operator Dm3 on Eq. (9), we haveDm3(RRDM(GYα;l,m))=(486rt+27r+299t)l3m3+(96rt-48r-48t)l3m2

After operating differential operator Dl3, we haveDl3Dm3(RRDM(GYα;l,m))=(13122rt+729r+8073t)l3m3+(2592rt-1296r-1296t)l3m2JDl3Dm3(RRDM(GYα;l,m))=(13122rt+729r+8073t)l6+(2592rt-1296r-1296t)l5Q-2JDl3Dm3(RRDM(GYα;l,m))=l-2[(13122rt+729r+8073t)l6+(2592rt-1296r-1296t)l5]Q-2JDl3Dm3(RRDM(GYα;l,m))=(13122rt+729r+8073t)l4+(2592rt-1296r-1296t)l3

After operating Integral operator Il3, we haveIl3Q-2JDl3Dm3(RRDM(GYα;l,m))=656132rt+72964r+807364tl4+(96rt-48r-48t)l3Il3Q-2JDl3Dm3(RRDM(GYα;l,m))|l=1=[656132rt+72964r+807364t)l4+(96rt-48r-48t)l3]|l=1RRDA(GYα)=963332rt-234364r+500164t

Reduced reverse degree based RRDM-polynomials for β-graphyne

Using Fig. 3, the reduced reverse degree-based edge division of α-graphdiyne is described below in Table 4.Figure 3 Structure of β-graphyne.

Table 4 Reduced reverse degree based edge partition of β-graphyne.

RRDE(λ,τ)	Cardinality	
E(3,3)	12rt+6r+18t	
E(3,2)	24rt-4r+4t	
E(2,2)	6rt-r+t	

The Reduce Reverse Degree-BasedM-Polynomial forGYβ(r,t)

Let GYβ(r,t) be graph of β-graphyne. Let RRD(i,j) be the set of all edges with a reduced Reverse degree of end vertices i, j and νi,j be the number of edges in RRD(i,j).RRD(i,j)={λτ∈E(G):RRD(λ)=i,RRD(τ)=j}RRD(3,3)={λτ∈E(G):RRD(λ)=3,RRD(τ)=3}RRD(3,2)={λτ∈E(G):RRD(λ)=3,RRD(τ)=2}RRD(2,2)={λτ∈E(G):RRD(λ)=2,RRD(τ)=2}

From Fig. 3 and Table 4, it is clear that

ν(3,3) = 12rt+6r+18t, ν(3,2) = 24rt-4r+4t , ν(2,2) = 6rt-r+t

The RRDM-Polynomial of GYβ(r,t) is obtained as followRRDM(GYβ)=∑i≤jν(i,j)limj=ν(3,3)l3m3+ν(3,2)l3m2+ν(2,2)l2m2

Putting values of ν(i,j), we obtain19 RRDM(GYβ;l,m)=(12rt+6r+18t)l3m3+(24rt-4r+4t)l3m2+(6rt-r+t)l2m2

The differential operators forGYβ(r,t)

In view of Table 1, using operators (3) and (4) along with Eq. (19), we get20 Dl(RRDM(GYβ;l,m))=(36rt+18r+54t)l3m3+(72rt-12r+12t)l3m2+(12rt-2r+2t)l2m2

21 Dm(RRDM(GYβ;l,m))=(36rt+18r+54t)l3m3+(48rt-8r+8t)l3m2+(12rt-2r+2t)l2m2

The integral operators forGYβ(r,t)

In view of Table 1, using operators (5) and (6) along with Eq. (19), we get22 Il(RRDM(GYβ;l,m))=(4rt+2r+6t)l3m3+8rt-43r+43tl3m2+3rt-12r+12tl2m2

23 Im(RRDM(GYβ;l,m))=(4rt+2r+6t)l3m3+(12rt-2r+2t)l3m2+3rt-12r+12tl2m2

Topological descriptors ofβ-Graphyne usingRRDM-Polynomial approach

Reduced Reverse 1st Zagreb Index ofGYβ(r,t)

In view of Table 1, adding Eqs. (20) and (21), we get(Dl+Dm)(RRDM(GYβ;l,m))=(36rt+18r+54t)l3m3+(72rt-12r+12t)l3m2+(12rt-2r+2t)l2m2+(36rt+18r+54t)l3m3+(48rt-8r+8t)l3m2+(12rt-2r+2t)l2m2(Dl+Dm)(RRDM(GYβ;l,m))|l=m=1=[(72rt+36r+108t)l3m3+(120rt-20r+20t)l3m2+(24rt-4r+4t)l2m2]|l=m=1RRDM1(GYβ)=216rt+12r+132t

Reduced Reverse 2nd Zagreb Index ofGYβ(r,t)

In view of Table 1, applying differential operator on Eq. (21), we haveDl[(Dm(RRDM(GYβ;l,m))]=Dl[(36rt+18r+54t)l3m3+(48rt-8r+8t)l3m2+(12rt-2r+2t)l2m2](DlDm)(RRDM(GYβ;l,m))|l=m=1=[(108rt+54r+162t)l3m3+(144rt-24r+24t)l3m2+(24rt-4r+4t)l2m2]|l=m=1RRDM2(GYβ)=276rt+26r+190t

Reduced Reverse Forgotten Index ofGYβ(r,t)

In view of Table 1, applying differential operators on Eqs. (20) and (21), we haveDl2(RRDM(GYβ;l,m))=(108rt+54r+162t)l3m3+(216rt-36r+36t)l3m2+(24rt-4r+4t)l2m2Dm2(RRDM(GYβ;l,m))=(108rt+54r+162t)l3m3+(96rt-16r+16t)l3m2+(24rt-4r+4t)l2m2(Dl2+Dm2)(M((Dl2+Dm2)(RRDM(GYβ;l,m))=(216rt+108r+324t)l3m3+(312rt-52r+52t)l3m2+(48rt-8r+8t)l2m2(Dl2+Dm2)(RRDM(GYβ;l,m))|l=m=1=(216rt+108r+324t)l3m3+(312rt-52r+52t)l3m2+(48rt-8r+8t)l2m2l3m2|l=m=1

After putting limits, we haveRRDF(GYβ)=576rt+48r+384t

Reduced Reverse Hyper 1st Zagreb Index ofGYβ(r,t)

In the view of Table 1, and after applying differential operator on Eqs. (20) and (21), we have(Dl2+Dm2)(RRDM(GYβ;l,m))=(216rt+108r+324t)l3m3+(312rt-52r+52t)l3m2+(48rt-8r+8t)l2m2(2DlDm)(RRDM(GYβ;l,m))=(216rt+108r+324t)l3m3+(288rt-48r+48t)l3m2+(48rt-8r+8t)l2m2(Dl+Dm)2(GYβ;l,m))|l=m=1=(432rt+216r+648t)l3m3+(600rt-100r+100t)l3m2+(96rt-16r+16t)l2m2|l=m=1

After putting limits, we haveRRDHM1(GYβ)=1128rt+100r+764t

Reduced Reverse Hyper 2nd Zagreb Index ofGYβ(r,t)

In the view of Table 1, and after applying differential operator on Eq. (21), we have(Dl2Dm2)(RRDM(GYβ;l,m))=Dl2[(108rt+54r+162t)l3m3+(96rt-16r+16t)l3m2+(24rt-4r+4t)l2m2](Dl2Dm2)(RRDM(GYβ;l,m))|l=m=1=(972rt+486r+1458t)l3m3+(864rt-144r+144t)l3m2+(96rt-16r+16t)l2m2|l=m=1

After putting limits, we haveRRDHM2(GYβ)=1932rt+326r+1618t

Reduced Reverse Sigma Index ofGYβ(r,t)

In the view Table 1, and after applying differential operator on Eqs. (20) and (21), we have(Dl2+Dm2)(RRDM(GYβ;l,m))=(216rt+108r+324t)l3m3+(312rt-52r+52t)l3m2+(48rt-8r+8t)l2m2(2DlDm)(RRDM(GYβ;l,m))=(216rt+108r+324t)l3m3+(288rt-48r+48t)l3m2+(48rt-8r+8t)l2m2(Dl-Dm)2(RRDM(GYβ;l,m))|l=m=1=(24rt-4r+4t)l3m2|l=m=1

After putting limits, we haveRRDσ(GYβ)=24rt-4r+4t

Reduced Reverse Second Modified Zagreb ofGYβ(r,t)

From Table 1, Second Modified Zagreb Index for RRDM-Polynomial is(IlIm)(RRDM(GYβ;l,m))=Il[Im(RRDM(GYβ;l,m)]=Il(4rt+2r+6t)l3m3+(12rt-2r+2t)l3m2+3rt-12r+12tl2m2

After operating integral operator Il (5), we have(IlIm)(RRDM(GYβ;l,m))|l=m=1=43rt+23r+2tl3m3+4rt-23r+23tl3m2+32rt-14r+14tl2m2|l=m=1

After putting limits, we haveRRDmM2(GYβ)=416rt-14r+3512t

Reduced Reverse Redefined Third Zagreb Index ofGYβ(r,t)

In view of Table 1, Reduced Reverse Redefined Third Zagreb Index for RRDM-Polynomial is computed as follow(Dl+Dm)(M(GYβ;l,m))=(72rt+36r+108t)l3m3+(120rt-20r+20t)l3m2+(24rt-4r+4t)l2m2Dm(Dl+Dm)(RRDM(GYβ;l,m)=(216rt+108r+324t)l3m3+(240rt-40r+40t)l3m2+(48rt-8r+8t)l2m2(DlDm)(Dl+Dm)(RRDM(GYβ;l,m))=(648rt+324r+972t)l3m3+(720rt-120r+120t)l3m2+(96rt-16r+16t)l2m2(DlDm)(Dl+Dm)(RRDM(GYβ;l,m))|l=m=1=(648rt+324r+972t)l3m3+(720rt-120r+120t)l3m2+(96rt-16r+16t)l2m2|l=m=1

After putting limits, we haveRRDReZG3(GYβ;r,t)=1464rt+188r+1108t

Reduced Reverse Symmetric Division Degree Index ofGYβ(r,t)

In view Table 1, using Eq. (23), we have(Im)(RRDM(GYβ;l,m))=(4rt+2r+6t)l3m3+(12rt-2r+2t)l3m2+3rt-12r+12tl2m2

After applying operator Dl (3), we have(DlIm)(RRDM(GYβ;l,m))=(12rt+6r+18t)l3m3+(36rt-6r+6t)l3m2+(6rt-r+t)l2m2

Also from Eq. (21), we getDm(RRDM(GYβ;l,m))=(36rt+18r+54t)l3m3+(48rt-8r+8t)l3m2+(12rt-2r+2t)l2m2

After operating integral operator Il (5), we haveIlDm(RRDM(GYβ;l,m))=(12rt+6r+18t)l3m3+16rt-83r+83tl3m2+(6rt-r+t)l2m2(DlIm+IlDm)(RRDM(GYβ;l,m))=(24rt+12r+36t)l3m3+52rt-263r+263tl3m2+(12rt-2r+2t)l2m2(DlIm+IlDm)(RRDM(GYβ;l,m))|l=m=1=(24rt+12r+36t)l3m3+52rt-263r+263tl3m2+(12rt-2r+2t)l2m2|l=m=1RRDSDD(GYβ)=88rt-43r+1403

Reduced Reverse Harmonic Index ofGYβ(r,t)

In view of Table 1, applying operator (8) on Eq. (19), we haveJRRDM(GYβ;l,m)=(12rt+6r+18t)l6+(24rt-4r+4t)l5+(6rt-r+t)l4

After operating integral operator Il (5), we haveIlJRRDM(GYβ;l,m)=(2rt+r+3t)l6+245rt-45r+45tl5+32rt-14r+14tl42IlJRRDM(GYβ;l,m)=(4rt+2r+6t)l6+485rt-85r+85tl5+3rt-12r+12tl42IlJRRDM(GYβ;l,m)|l=m=1=(4rt+2r+6t)l6+485rt-85r+85tl5+3rt-12r+12tl4|l=1RRDH(GYβ)=835rt-110r+8110t

Reduced Reverse Inverse Sum Index ofGYβ(r,t)

From Table 1, Harmonic Index for RRDM-Polynomial isDlDm(RRDM(GYβ;l,m))=(108rt+54r+162t)l3m3+(144rt-24r+24t)l3m2+(24rt-4r+4t)l2m2JDlDm(RRDM(GYβ;l,m))=(108rt+54r+162t)l6+(144rt-24r+24t)l5+(24rt-4r+4t)l4

After operating integral operator Il (5), we haveIlJDlDm(RRDM(GYβ;l,m))=(18rt+9r+27t)l6+1445rt-245r+245tl5+(6rt-r+t)l4IlJDlDm(RRDM(GYβ;l,m))|l=1=(18rt+9r+27t)l6+1445rt-245r+245tl5+(6rt-r+t)l4|l=1RRDI(GYβ)=2645rt+165r+1645t

Reduced Reverse Augmented Index ofGYβ(r,t)

In view Table 1, applying operator Dm3 on Eq. (19), we haveDm3(RRDM(GYβ;l,m))=(324rt+162r+486t)l3m3+(192rt-32r+32t)l3m2+(48rt-8r+8t)l2m2

After operating differential operator Dl3, we haveDl3Dm3(RRDM(GYβ;l,m))=(8748rt+4374r+13122t)l3m3+(5184rt-864r+864t)l3m2+(384rt-64r+64t)l2m2JDl3Dm3(RRDM(GYβ;l,m))=(8748rt+4374r+13122t)l6+(5184rt-864r+864t)l5+(384rt-64r+64t)l4Q-2JDl3Dm3(RRDM(GYβ;l,m))=l-2[(8748rt+4374r+13122t)l6+(5184rt-864r+864t)l5+(384rt-64r+64t)l4]Q-2JDl3Dm3(RRDM(GYβ;l,m))=(8748rt+4374r+13122t)l4+(5184rt-864r+864t)l3+(384rt-64r+64t)l2

After operating integral operator Il3, we haveIl3Q-2JDl3Dm3(RRDM(GYβ;l,m))=218716rt+218732r+656132tl4+(192rt-32r+32t)l3+(48rt-8r+8t)l2Il3Q-2JDl3Dm3(RRDM(GYβ;l,m))|l=1=[218716rt+218732r+656132t)l4+(192rt-32r+32t)l3+(48rt-8r+8t)l2]|l=1RRDA(GYβ)=602716rt+90732r+784132t

Prediction of the features of graphene nanoribbons

This section describes one of the main focuses of this work, emphasizing the role of topological descriptors in QSPR study and giving instances of its elements of assessment and prediction for boron sheets. An equation derived via regression analysis demonstrated a relationship between topological descriptors and important characteristics of boron sheets. Topological descriptors in QSPR use mathematical models to characterize a Molecular Graph’s properties or functions. Molecular descriptors and the physical properties of chemical compounds are linked by QSPR.

To classify and assess a substance’s properties, a clear and concise strategy must be developed69. Consequently, it is critical to investigate and understand the structural properties of molecular compounds. The properties of molecular compounds are analyzed or predicted using modeling techniques or methodologies such as exponential, logarithmic, cubic, linear, and quadratic regression analysis70–73. We provide a quadratic regression analysis of the graphene. Apart from its structural and electrical stability, graphene nanoribbons exhibit interesting features related to chemical bonding. Regression analysis was used to examine two-dimensional graphene nanoribbons, such as the α-graphyne nanoribbon and the α-graphdiyne nanoribbon74 and β-graphyne75. Figures 1, 2 and 3 shows an illustration of the graphyne and graphdiyne nanoribbons discussed above. Table 6 provides the reduced reverse degree-vertex value of the graphene nanoribbon’s structure.

In the present research, we examine the graphyne and graphdiyne nanoribbon’s characteristics such as Poisson’s ratio PR and Young’s modulus YM, which can be determined using the elastic constant. Table 5 summarizes the results for Young’s modulus YM, and Poisson’s ratio PR of different graphyne and graphdiyne nanoribbon’s. The Young’s Modulus show a material’s resistance to changes in length when subjected to tension or compression and Poisson’s ratio measures the deformation (expansion or contraction) of a material perpendicular to the direction of loading.Table 5 Experimental values of Young’s modulus and Poisson’s ratio of graphene nanoribbons.

Graphene nanoribbons	Young’s modulus YM	Poisson’s ratio PR	
α-Graphyne	42.8	0.72	
α-Graphdiyne	14.1	0.85	
β-Graphyne	93.6	0.52	

Table 6 Computaional values of RRD descriptors of graphene and graphdiyne.

TD′s	α-Graphyne	α-Graphdiyne	β-Graphyne	
RRDM1	900	1548	2376	
RRDM2	1242	2214	3132	
RRDF	2556	4500	6480	
RRDHM1	5040	8928	12,744	
RRDHM2	9882	18,630	23,220	
RRDσ	72	72	216	
RRDmM2	22	34	69.5	
RRDReZG3	7020	12,852	17,064	
RRDSDD	426	552	834.66	
RRDH	58.8	102.4	173.4	
RRDI	221.4	383.4	583.2	
RRDA	1601.16	2677.59	4210.31	

Properties analysis through QSPR modeling

Regression modeling is used to examine the mechanical characteristics, Young’s modulus, and Poisson’s ratio of the mentioned graphene derivative’s using topological descriptors. Legendre et al.76 and Gauss et al.77 established the least squares approach to linear and quadratic regression in 1805 and 1809, respectively. Regression analysis is a statistical technique that determines the correlation between many variables. The correlation coefficient ranges from − 1 to 1. Perfect positive and negative correlation are 1 and − 1, respectively, while near-zero correlation implies inadequate correlation. The equation relating the characteristics and descriptors is derived using regression analysis and a correlation coefficient.PCP=p(TD)2+n(TD)+m

where PCP is the physiochemical property of graphene, TD is the topological descriptor, m is the invariant, p and n is the regression coefficient. The correlation coefficient of different topological descriptors has been studied for both chemical attributes. We have concluded that Reduced Reverse degree RRD Hyper 2nd Zagrab Index and Reduced Reverse degree RRD Redefined Third Zagreb Index has a strong correlation for Young’s modulus and Poison’s ratio respectively. The quadratic regression equations for Young’s modulus are shown as follows:YM=2×10-6(RRDHM2)2-0.0473(RRDHM2)+359.57YM=2×10-6(RRDReZG3)2-0.052(RRDReZG3)+291.09

where YM is the Young’s modulus and RRDHM2 is the Reduced Reverse Degree Hyper 2nd Zagrab Index. Similarly, the quadratic regression equations for Poisson’s Ratio are determined as follows:PR=-7×10-9(RRDHM2)2+0.0002(RRDHM2)-0.6243PR=-1×10-8(RRDReZG3)2+0.0002(RRDReZG3)-0.3405

where PR is the Poison’s Ratio and RRDReZG3 is the Reduced Reverse degree Refefined Third Zagreb Index. Suitable regression models can be used to forecast the molecular features that have a larger dimension. In Fig. 4, the scatter plots corresponding to the most highly associated properties and descriptors are displayed.Figure 4 Scattering-based visualization of properties and descriptors.

Conclusion

The concept of a RRDM-polynomial based on reduced reverse degrees of a graph is introduced in this paper, and from this, differential and integral operators of a graph are extracted which are helpful in the formulation of reverse degree-based TDs. We have formulated twelve reverse degree-based TDs as represented in Table 6 through this methodology. For the structure of the α-graphyne, β-graphyne and α-graphdiyne, this study presented the following RRDM-Polynomial.RRDM(GYα;l,m)=(6rt+3r+9t)l3m3+(12rt-6r-6t)l3m2RRDM(GDα;l,m)=(18rt+r+11t)l3m3+(12rt-6r-6t)l3m2RRDM(GYβ;l,m)=(12rt+6r+18t)l3m3+(24rt-4r+4t)l3m2+(6rt-r+t)l2m2

Differential OperatorsDl(RRDM(GYα;l,m))=(18rt+9r+27t)l3m3+(36rt-18r-18t)l3m2Dm(RRDM(GYα;l,m))=(18rt+9r+27t)l3m3+(24rt-12r-12t)l3m2Dl(RRDM(GDα;l,m))=(54rt+3r+33t)l3m3+(36rt-18r-18t)l3m2Dm(RRDM(GDα;l,m))=(54rt+3r+33t)l3m3+(24rt-12r-12t)l3m2Dl(RRDM(GYβ;l,m))=(36rt+18r+54t)l3m3+(72rt-12r+12t)l3m2+(12rt-2r+2t)l2m2Dm(RRDM(GYβ;l,m))=(36rt+18r+54t)l3m3+(48rt-8r+8t)l3m2+(12rt-2r+2t)l2m2

Integral OperatorsIl(RRDM(GYα;l,m))=(2rt+r+3t)l3m3+(4rt-2r-2t)l3m2Im(RRDM(GYα;l,m))=(2rt+r+3t)l3m3+(6rt-3r-3t)l3m2Il(RRDM(GDα;l,m))=6rt+13r+113tl3m3+(4rt-3r-3t)l3m2Im(RRDM(GDα;l,m))=6rt+13r+113tl3m3+(6rt-3r-3t)l3m2Il(RRDM(GYβ;l,m))=(4rt+2r+6t)l3m3+8rt-43r+43tl3m2+3rt-12r+12tl2m2Im(RRDM(GYβ;l,m))=(4rt+2r+6t)l3m3+(12rt-2r+2t)l3m2+3rt-12r+12tl2m2

We also estimated the physicochemical properties of the α-graphyne, β-graphyne and α-graphdiyne by employing the computed reduce reverse degree-based topological descriptors and find the best estimations for the Poisson’s Ratio and Young’s Modulus of the graphene and its derivatives through RRDHM2 and RRDReZG3.

Acknowledgements

The authors appreciated the kind support from the researchers Supporting Project Number (RSP2024R440), King Saud University, Riyadh, Saudi Arabia.

Author contributions

Abdul Rauf Khan contributed to the Investigation, analyzing the data curation, and designing the experiments. Saad Amin Bhatti contributed to data analysis, computation, funding resources, and calculation verifications. Ferdous Tawfiq contributed to the computation and investigated and approved the final draft of the paper. Muhammad Kamran Siddiqui contributed to supervision, conceptualization, and Methodology. Shahid Hussain contributed to Matlab calculations, Maple graphs improvement project administration, and wrote the initial draft of the paper. Mustafa Ahmed Ali contributes to formal analyzing experiments, software, validation, and funding. All authors read and approved the final version.

Data availibility

All data generated or analyzed during this study are included in this article.

Declarations

Competing interests

The authors declare that they have no competing interests.

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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References

1. Baughman RH Eckhardt H Kertesz M Structure-property predictions for new planar forms of carbon: Layered phases containing sp 2 and sp atoms J. Chem. Phys. 1987 87 11 6687 6699 10.1063/1.453405
Baughman, R. H., Eckhardt, H. & Kertesz, M. Structure-property predictions for new planar forms of carbon: Layered phases containing sp 2 and sp atoms. J. Chem. Phys. 87(11), 6687–6699 (1987).10.1063/1.453405
2. Autreto PAS De Sousa JM Galvao DS Site-dependent hydrogenation on graphdiyne Carbon 2014 77 829 834 10.1016/j.carbon.2014.05.088
Autreto, P. A. S., De Sousa, J. M. & Galvao, D. S. Site-dependent hydrogenation on graphdiyne. Carbon 77, 829–834 (2014).10.1016/j.carbon.2014.05.088
3. Felix LC Woellner CF Galvao DS Mechanical and energy-absorption properties of schwarzites Carbon 2020 157 670 680 10.1016/j.carbon.2019.10.066
Felix, L. C., Woellner, C. F. & Galvao, D. S. Mechanical and energy-absorption properties of schwarzites. Carbon 157, 670–680 (2020).10.1016/j.carbon.2019.10.066
4. Coluci VR Braga SF Legoas SB Galvao DS Baughman RH Families of carbon nanotubes: Graphyne-based nanotubes Phys. Rev. B 2003 68 3 035430 10.1103/PhysRevB.68.035430
Coluci, V. R., Braga, S. F., Legoas, S. B., Galvao, D. S. & Baughman, R. H. Families of carbon nanotubes: Graphyne-based nanotubes. Phys. Rev. B 68(3), 035430 (2003).10.1103/PhysRevB.68.035430
5. Chen, S., Semenov, I., Zhang, F., Yang, Y., Geng, J., Feng, X., Lei, K. An effective framework for predicting drug-drug interactions based on molecular substructures and knowledge graph neural network. Comput. Biol. Med. 169, 107900 (2024).
6. Malko D Neiss C Vines F Görling A Competition for graphene: graphynes with direction-dependent dirac cones Phys. Rev. Lett. 2012 108 8 086804 10.1103/PhysRevLett.108.086804 22463556
Malko, D., Neiss, C., Vines, F. & Görling, A. Competition for graphene: graphynes with direction-dependent dirac cones. Phys. Rev. Lett. 108(8), 086804 (2012).22463556 10.1103/PhysRevLett.108.086804
7. Jia Z Li Y Zuo Z Liu H Huang C Li Y Synthesis and Properties of 2D carbon graphdiyne Acc. Chem. Res. 2017 50 10 2470 2478 10.1021/acs.accounts.7b00205 28915007
Jia, Z. et al. Synthesis and Properties of 2D carbon graphdiyne. Acc. Chem. Res. 50(10), 2470–2478 (2017).28915007 10.1021/acs.accounts.7b00205
8. Baughman RH Eckhardt H Kertesz M Structure-property predictions for new planar forms of carbon: Layered phases containing sp 2 and sp atoms J. Chem. Phys. 1987 87 11 6687 6699 10.1063/1.453405
Baughman, R. H., Eckhardt, H. & Kertesz, M. Structure-property predictions for new planar forms of carbon: Layered phases containing sp 2 and sp atoms. J. Chem. Phys. 87(11), 6687–6699 (1987).10.1063/1.453405
9. Habib MR Liang T Yu X Pi X Liu Y Xu M A review of theoretical study of graphene chemical vapor deposition synthesis on metals: nucleation, growth, and the role of hydrogen and oxygen Rep. Prog. Phys. 2018 81 3 036501 10.1088/1361-6633/aa9bbf 29355108
Habib, M. R. et al. A review of theoretical study of graphene chemical vapor deposition synthesis on metals: nucleation, growth, and the role of hydrogen and oxygen. Rep. Prog. Phys. 81(3), 036501 (2018).29355108 10.1088/1361-6633/aa9bbf
10. Randviir EP Brownson DA Banks CE A decade of graphene research: Production, applications and outlook Mater. Today 2014 17 9 426 432 10.1016/j.mattod.2014.06.001
Randviir, E. P., Brownson, D. A. & Banks, C. E. A decade of graphene research: Production, applications and outlook. Mater. Today 17(9), 426–432 (2014).10.1016/j.mattod.2014.06.001
11. Coroş M Pogăcean F Măgeruşan L Socaci C Pruneanu S A brief overview on synthesis and applications of graphene and graphene-based nanomaterials Front. Mater. Sci. 2019 13 23 32 10.1007/s11706-019-0452-5
Coroş, M., Pogăcean, F., Măgeruşan, L., Socaci, C. & Pruneanu, S. A brief overview on synthesis and applications of graphene and graphene-based nanomaterials. Front. Mater. Sci. 13, 23–32 (2019).10.1007/s11706-019-0452-5
12. Çiftçi, İ., Ediz, S., Aldemir, M. & Yamaç, K. On R, S and Van entropies of beta graphene. Graphs Linear Algebra 2023(1), (2023).
13. Majidi R Structural and electronic properties of α2-graphyne nanotubes: A density functional theory study J. Electron. Mater. 2018 47 5 2890 2896 10.1007/s11664-018-6156-2
Majidi, R. Structural and electronic properties of 2-graphyne nanotubes: A density functional theory study. J. Electron. Mater. 47(5), 2890–2896 (2018).10.1007/s11664-018-6156-2
14. Xu, X., Fu, X., Zhao, H., Liu, M., Xu, A., & Ma, Y. Three-dimensional reconstruction and geometric morphology analysis of lunar small craters within the patrol range of the Yutu-2 Rover. Remote Sens. 15(17), 4251 (2023).
15. Yu H Xue Y Li Y Graphdiyne and its assembly architectures: Synthesis, functionalization, and applications Adv. Mater. 2019 31 42 1803101 10.1002/adma.201803101
Yu, H., Xue, Y. & Li, Y. Graphdiyne and its assembly architectures: Synthesis, functionalization, and applications. Adv. Mater. 31(42), 1803101 (2019).10.1002/adma.201803101
16. Couto, R. & Silvestre, N. Finite element modelling and mechanical characterization of graphyne. J. Nanomater. (2016).
17. Arockiaraj M Clement J Tratnik N Mushtaq S Balasubramanian K Weighted Mostar descriptors as measures of molecular peripheral shapes with applications to graphene, graphyne and graphdiyne nanoribbons SAR QSAR Environ. Res. 2020 31 3 187 208 10.1080/1062936X.2019.1708459 31960721
Arockiaraj, M., Clement, J., Tratnik, N., Mushtaq, S. & Balasubramanian, K. Weighted Mostar descriptors as measures of molecular peripheral shapes with applications to graphene, graphyne and graphdiyne nanoribbons. SAR QSAR Environ. Res. 31(3), 187–208 (2020).31960721 10.1080/1062936X.2019.1708459
18. Rada J Vertex-degree based topological descriptors of graphene Polycyclic Aromat. Compd. 2022 42 4 1524 1532 10.1080/10406638.2020.1785897
Rada, J. Vertex-degree based topological descriptors of graphene. Polycyclic Aromat. Compd. 42(4), 1524–1532 (2022).10.1080/10406638.2020.1785897
19. Arockiaraj M Klavžar S Mushtaq S Balasubramanian K Topological characterization of the full k-subdivision of a family of partial cubes and their applications to α-types of novel graphyne and graphdiyne materials Polycyclic Aromat. Compd. 2021 41 9 1902 1924 10.1080/10406638.2019.1703766
Arockiaraj, M., Klavžar, S., Mushtaq, S. & Balasubramanian, K. Topological characterization of the full k-subdivision of a family of partial cubes and their applications to -types of novel graphyne and graphdiyne materials. Polycyclic Aromat. Compd. 41(9), 1902–1924 (2021).10.1080/10406638.2019.1703766
20. Chu YM Khan AR Ghani MU Ghaffar A Inc M Computation of zagreb polynomials and zagreb descriptors for benzenoid triangular and hourglass system Polycyclic Aromat. Compd. 2023 43 5 4386 4395 10.1080/10406638.2022.2090970
Chu, Y. M., Khan, A. R., Ghani, M. U., Ghaffar, A. & Inc, M. Computation of zagreb polynomials and zagreb descriptors for benzenoid triangular and hourglass system. Polycyclic Aromat. Compd. 43(5), 4386–4395 (2023).10.1080/10406638.2022.2090970
21. Chen, J., Song, Y., Li, D., Lin, X., Zhou, S., Xu, W. Specular removal of industrial metal objects without changing lighting configuration. IEEE Trans. Ind. Inf. 20(3), 3144–3153 (2024).
22. Xu H Li Q Chen J Highlight removal from a single grayscale image using attentive GAN Appl. Artif. Intell. 2022 36 1 1988441 10.1080/08839514.2021.1988441
Xu, H., Li, Q. & Chen, J. Highlight removal from a single grayscale image using attentive GAN. Appl. Artif. Intell. 36(1), 1988441 (2022).10.1080/08839514.2021.1988441
23. Madurani KA Suprapto S Machrita NI Bahar SL Illiya W Kurniawan F Progress in graphene synthesis and its application: history, challenge and the future outlook for research and industry ECS J. Solid State Sci. Technol. 2020 9 9 093013 10.1149/2162-8777/abbb6f
Madurani, K. A. et al. Progress in graphene synthesis and its application: history, challenge and the future outlook for research and industry. ECS J. Solid State Sci. Technol. 9(9), 093013 (2020).10.1149/2162-8777/abbb6f
24. Ahmad A Computation of certain topological properties of para-line graph of honeycomb networks and graphene Disc. Math. Algorithms Appl. 2017 9 05 1750064 10.1142/S1793830917500641
Ahmad, A. Computation of certain topological properties of para-line graph of honeycomb networks and graphene. Disc. Math. Algorithms Appl. 9(05), 1750064 (2017).10.1142/S1793830917500641
25. Ghani MU Sultan F Tag El Din ESM Khan AR Liu JB Cancan M A paradigmatic approach to find the valency-based K-Banhatti and redefined zagreb entropy for niobium oxide and a metal-organic framework Molecules 2022 27 20 6975 10.3390/molecules27206975 36296567
Ghani, M. U. et al. A paradigmatic approach to find the valency-based K-Banhatti and redefined zagreb entropy for niobium oxide and a metal-organic framework. Molecules 27(20), 6975 (2022).36296567 10.3390/molecules27206975
26. Liu JB Gu JJ Wang K The expected values for the Gutman index, Schultz index, and some Sombor descriptors of a random cyclooctane chain Int. J. Quant. Chem. 2023 123 3 e27022 10.1002/qua.27022
Liu, J. B., Gu, J. J. & Wang, K. The expected values for the Gutman index, Schultz index, and some Sombor descriptors of a random cyclooctane chain. Int. J. Quant. Chem. 123(3), e27022 (2023).10.1002/qua.27022
27. Jia Y Yu W Chen G Zhao L Nighttime road scene image enhancement based on cycle-consistent generative adversarial network Sci. Rep. 2024 14 1 14375 10.1038/s41598-024-65270-3 38909068
Jia, Y., Yu, W., Chen, G. & Zhao, L. Nighttime road scene image enhancement based on cycle-consistent generative adversarial network. Sci. Rep. 14(1), 14375 (2024).38909068 10.1038/s41598-024-65270-3
28. Amigó JM Gálvez J Villar VM A review on molecular topology: Applying graph theory to drug discovery and design Naturwissenschaften 2009 96 749 761 10.1007/s00114-009-0536-7 19513596
Amigó, J. M., Gálvez, J. & Villar, V. M. A review on molecular topology: Applying graph theory to drug discovery and design. Naturwissenschaften 96, 749–761 (2009).19513596 10.1007/s00114-009-0536-7
29. Chen S Semenov I Zhang F Yang Y Geng J Feng X Lei K An effective framework for predicting drug-drug interactions based on molecular substructures and knowledge graph neural network Comput. Biol. Med. 2024 169 107900 10.1016/j.compbiomed.2023.107900 38199213
Chen, S. et al. An effective framework for predicting drug-drug interactions based on molecular substructures and knowledge graph neural network. Comput. Biol. Med. 169, 107900 (2024).38199213 10.1016/j.compbiomed.2023.107900
30. Edition, S. & Rosen, K. H. Discrete mathematics and its applications.
31. Wiener H Structural determination of paraffin boiling points J. Am. Chem. Soc. 1947 69 1 17 20 10.1021/ja01193a005 20291038
Wiener, H. Structural determination of paraffin boiling points. J. Am. Chem. Soc. 69(1), 17–20 (1947).20291038 10.1021/ja01193a005
32. Yan A Liu R Cui J Ni T Girard P Wen X Zhang J Designs of BCD adder based on excess-3 code in quantum-dot cellular automata IEEE Trans. Circuits Syst. II Express Briefs 2023 70 6 2256 2260
Yan, A. et al. Designs of BCD adder based on excess-3 code in quantum-dot cellular automata. IEEE Trans. Circuits Syst. II Express Briefs 70(6), 2256–2260 (2023).
33. Khan, A. R., Awan, N. U. H., Tchier, F., Alahmari, S. D., Khalel, A. F. & Hussain, S. An estimation of physiochemical properties of bladder cancer drugs via degree-based chemical bonding topological descriptors. J. BioMolecular Graph Dyn. 1–9 (2023).
34. Khan, A. R., Zia, A., Campeña, F. J. H., Siddiqui, M. K., Tchier, F. & Hussain, S. Investigations of entropy double and strong double graph of silicon carbide. Silicon 1–11 (2024).
35. Hayat, S., Khan, S., Khan, A. & Imran, M. Distance-based topological descriptors for measuring the -electronic energy of benzenoid hydrocarbons with applications to carbon nanotubes. Math. Methods Appl. Sci. (2020).
36. Hayat S Khan S Khan A Imran M A computer-based method to determine predictive potential of distance-spectral descriptors for measuring the π-electronic energy of benzenoid hydrocarbons with applications IEEE Access 2021 9 19238 19253 10.1109/ACCESS.2021.3053270
Hayat, S., Khan, S., Khan, A. & Imran, M. A computer-based method to determine predictive potential of distance-spectral descriptors for measuring the -electronic energy of benzenoid hydrocarbons with applications. IEEE Access 9, 19238–19253 (2021).10.1109/ACCESS.2021.3053270
37. Hayat S Khan S Khan A Liu JB Valency-based molecular descriptors for measuring the π-electronic energy of lower polycyclic aromatic hydrocarbons Polycyclic Aromat. Compd. 2022 42 4 1113 1129 10.1080/10406638.2020.1768414
Hayat, S., Khan, S., Khan, A. & Liu, J. B. Valency-based molecular descriptors for measuring the -electronic energy of lower polycyclic aromatic hydrocarbons. Polycyclic Aromat. Compd. 42(4), 1113–1129 (2022).10.1080/10406638.2020.1768414
38. Khan S Comparative study of domination parameters with the π-electronic energy of benzenoid hydrocarbons Int. J. Quant. Chem. 2023 123 20 e27192 10.1002/qua.27192
Khan, S. Comparative study of domination parameters with the -electronic energy of benzenoid hydrocarbons. Int. J. Quant. Chem. 123(20), e27192 (2023).10.1002/qua.27192
39. Malik, M. Y. H., Hayat, S., Khan, S. & Binyamin, M. A. Predictive potential of spectrum-based topological descriptors for measuring the -electronic energy of benzenoid hydrocarbons with applications to boron triangular and boron -nanotubes. Math. Methods Appl. Sci (2021).
40. Zhao C Tang X Zhao J Cao J Jiang Z Qin J MOF derived core-shell CuO/C with temperature-controlled oxygen-vacancy for real time analysis of glucose J. Nanobiotechnol. 2022 20 1 507 10.1186/s12951-022-01715-z
Zhao, C. et al. MOF derived core-shell CuO/C with temperature-controlled oxygen-vacancy for real time analysis of glucose. J. Nanobiotechnol. 20(1), 507 (2022).10.1186/s12951-022-01715-z
41. Doley A Buragohain J Bharali A Inverse sum index status index of graphs and its applications to octane isomers and benzenoid hydrocarbons Chemom. Intell. Lab. Syst. 2020 203 104059 10.1016/j.chemolab.2020.104059
Doley, A., Buragohain, J. & Bharali, A. Inverse sum index status index of graphs and its applications to octane isomers and benzenoid hydrocarbons. Chemom. Intell. Lab. Syst. 203, 104059 (2020).10.1016/j.chemolab.2020.104059
42. Khan AR Awan NUH Ghani MU Eldin SM Karamti H Jawhari AH Mukhrish YE Fundamental aspects of skin cancer drugs via degree-based chemical bonding topological descriptors Molecules 2023 28 9 368 10.3390/molecules28093684 36615562
Khan, A. R. et al. Fundamental aspects of skin cancer drugs via degree-based chemical bonding topological descriptors. Molecules 28(9), 368 (2023).36615562 10.3390/molecules28093684
43. Imran M Khan AR Husin MN Tchier F Ghani MU Hussain S Computation of entropy measures for metal-organic frameworks Molecules 2023 28 12 4726 10.3390/molecules28124726 37375281
Imran, M. et al. Computation of entropy measures for metal-organic frameworks. Molecules 28(12), 4726 (2023).37375281 10.3390/molecules28124726
44. Hayat S Khan S Quality testing of spectrum-based valency descriptors for polycyclic aromatic hydrocarbons with applications J. Mol. Struct. 2021 1228 129789 10.1016/j.molstruc.2020.129789
Hayat, S. & Khan, S. Quality testing of spectrum-based valency descriptors for polycyclic aromatic hydrocarbons with applications. J. Mol. Struct. 1228, 129789 (2021).10.1016/j.molstruc.2020.129789
45. Hayat S Khan S Imran M Quality testing of spectrum-based distance descriptors for polycyclic aromatic hydrocarbons with applications to carbon nanotubes and nanocones Arab. J. Chem. 2021 14 3 102994 10.1016/j.arabjc.2021.102994
Hayat, S., Khan, S. & Imran, M. Quality testing of spectrum-based distance descriptors for polycyclic aromatic hydrocarbons with applications to carbon nanotubes and nanocones. Arab. J. Chem. 14(3), 102994 (2021).10.1016/j.arabjc.2021.102994
46. Hayat S Khan S Imran M Liu JB Quality testing of distance-based molecular descriptors for benzenoid hydrocarbons J. Mol. Struct. 2020 1222 128927 10.1016/j.molstruc.2020.128927
Hayat, S., Khan, S., Imran, M. & Liu, J. B. Quality testing of distance-based molecular descriptors for benzenoid hydrocarbons. J. Mol. Struct. 1222, 128927 (2020).10.1016/j.molstruc.2020.128927
47. Chen, Q., Yang, L., Zhao, Y., Wang, Y., Zhou, H., & Chen, X. Shortest path in LEO satellite constellation networks: An explicit analytic approach. IEEE J. Select. Areas Commun. 42(5), 1175–1187 (2024).
48. Zaman S Ahmed W Sakeena A Rasool KB Ashebo MA Mathematical modeling and topological graph description of dominating David derived networks based on edge partitions Sci. Rep. 2023 13 1 15159 10.1038/s41598-023-42340-6 37704710
Zaman, S., Ahmed, W., Sakeena, A., Rasool, K. B. & Ashebo, M. A. Mathematical modeling and topological graph description of dominating David derived networks based on edge partitions. Sci. Rep. 13(1), 15159 (2023).37704710 10.1038/s41598-023-42340-6
49. Naeem M Iqbal Z Maqbool S Qureshi TM Ve-degree and Ev-degree based topological properties of magnesium oxide MgO (111) structures Front. Chem. Sci. 2022 3 1 45 55 10.52700/fcs.v3i1.39
Naeem, M., Iqbal, Z., Maqbool, S. & Qureshi, T. M. Ve-degree and Ev-degree based topological properties of magnesium oxide MgO (111) structures. Front. Chem. Sci. 3(1), 45–55 (2022).10.52700/fcs.v3i1.39
50. Khan AR Ghani MU Ghaffar A Asif HM Inc M Characterization of temperature descriptors of silicates SILICON 2023 15 15 6533 6539 10.1007/s12633-023-02298-6
Khan, A. R., Ghani, M. U., Ghaffar, A., Asif, H. M. & Inc, M. Characterization of temperature descriptors of silicates. SILICON 15(15), 6533–6539 (2023).10.1007/s12633-023-02298-6
51. Eryaşar E Sözen EÖ Büyükköse Ş New formulas and new bounds for the first and second zagreb descriptors of phenylenes Karadeniz Fen Bilimleri Dergisi 2024 14 2 468 475 10.31466/kfbd.1362864
Eryaşar, E., Sözen, E. Ö. & Büyükköse, Ş. New formulas and new bounds for the first and second zagreb descriptors of phenylenes. Karadeniz Fen Bilimleri Dergisi 14(2), 468–475 (2024).10.31466/kfbd.1362864
52. Guo S Wang S Twisted relative Rota-Baxter operators on Leibniz conformal algebras Comm. Algebra 2024 52 9 3946 3959 10.1080/00927872.2024.2337276
Guo, S. & Wang, S. Twisted relative Rota-Baxter operators on Leibniz conformal algebras. Comm. Algebra 52(9), 3946–3959 (2024).10.1080/00927872.2024.2337276
53. Öztürk Sözen E Eryaşar E An algebraic approach to calculate some topological codescriptors and QSPR analysis of some novel drugs used in the treatment of breast cancer Polycyclic Aromat. Compd. 2024 44 4 2226 2243 10.1080/10406638.2023.2214286
Öztürk Sözen, E. & Eryaşar, E. An algebraic approach to calculate some topological codescriptors and QSPR analysis of some novel drugs used in the treatment of breast cancer. Polycyclic Aromat. Compd. 44(4), 2226–2243 (2024).10.1080/10406638.2023.2214286
54. Öztürk Sözen E Eryaşar E QSPR analysis of some drug candidates investigated for COVID-19 via new topological codescriptors Polycyclic Aromat. Compd. 2024 44 2 1291 1308 10.1080/10406638.2023.2191974
Öztürk Sözen, E. & Eryaşar, E. QSPR analysis of some drug candidates investigated for COVID-19 via new topological codescriptors. Polycyclic Aromat. Compd. 44(2), 1291–1308 (2024).10.1080/10406638.2023.2191974
55. West DB Introduction to graph theory 2001 Upper Saddle River Prentice hall
West, D. B. Introduction to graph theory (Prentice hall, Upper Saddle River, 2001).
56. Furtula B Das KC Gutman I Comparative analysis of symmetric division deg index as potentially useful molecular descriptor Int. J. Quantum Chem. 2018 118 17 e25659 10.1002/qua.25659
Furtula, B., Das, K. C. & Gutman, I. Comparative analysis of symmetric division deg index as potentially useful molecular descriptor. Int. J. Quantum Chem. 118(17), e25659 (2018).10.1002/qua.25659
57. Hosamani SM Correlation of domination parameters with physicochemical properties of octane isomers Appl. Math. Nonlinear Sci. 2016 1 2 345 352 10.21042/AMNS.2016.2.00029
Hosamani, S. M. Correlation of domination parameters with physicochemical properties of octane isomers. Appl. Math. Nonlinear Sci. 1(2), 345–352 (2016).10.21042/AMNS.2016.2.00029
58. Zaman S Hakami KH Rasheed S Agama FT Reduced reverse degree-based topological descriptors of graphyne and graphdiyne nanoribbons with applications in chemical analysis Sci. Rep. 2024 14 1 547 10.1038/s41598-023-51112-1 38177204
Zaman, S., Hakami, K. H., Rasheed, S. & Agama, F. T. Reduced reverse degree-based topological descriptors of graphyne and graphdiyne nanoribbons with applications in chemical analysis. Sci. Rep. 14(1), 547 (2024).38177204 10.1038/s41598-023-51112-1
59. Coluci VR Braga SF Legoas SB Galvao DS Baughman RH Families of carbon nanotubes: Graphyne-based nanotubes Phys. Rev. B 2003 68 3 035430 10.1103/PhysRevB.68.035430
Coluci, V. R., Braga, S. F., Legoas, S. B., Galvao, D. S. & Baughman, R. H. Families of carbon nanotubes: Graphyne-based nanotubes. Phys. Rev. B 68(3), 035430 (2003).10.1103/PhysRevB.68.035430
60. Haley MM Brand SC Pak JJ Carbon networks based on dehydrobenzoannulenes: Synthesis of graphdiyne substructures Angew. Chem. Int. Ed. Engl. 1997 36 8 836 838 10.1002/anie.199708361
Haley, M. M., Brand, S. C. & Pak, J. J. Carbon networks based on dehydrobenzoannulenes: Synthesis of graphdiyne substructures. Angew. Chem. Int. Ed. Engl. 36(8), 836–838 (1997).10.1002/anie.199708361
61. Ivanovskii AL Graphynes and graphdyines Prog. Solid State Chem. 2013 41 1–2 1 19 10.1016/j.progsolidstchem.2012.12.001
Ivanovskii, A. L. Graphynes and graphdyines. Prog. Solid State Chem. 41(1–2), 1–19 (2013).10.1016/j.progsolidstchem.2012.12.001
62. Pei Y Mechanical properties of graphdiyne sheet Phys. B 2012 407 22 4436 4439 10.1016/j.physb.2012.07.026
Pei, Y. Mechanical properties of graphdiyne sheet. Phys. B 407(22), 4436–4439 (2012).10.1016/j.physb.2012.07.026
63. Alsaadi FE Salman M Rehman MU Khan AR Cao J Alassafi MO On the geodesic identification of vertices in convex plane graphs Math. Probl. Eng. 2020 2020 1 13 10.1155/2020/7483291
Alsaadi, F. E. et al. On the geodesic identification of vertices in convex plane graphs. Math. Probl. Eng. 2020, 1–13 (2020).10.1155/2020/7483291
64. Kulli VR Reverse Zagreb and reverse hyper-Zagreb indices and their polynomials of rhombus silicate networks Ann. Pure Appl. Math. 2018 16 1 47 51 10.22457/apam.v16n1a6
Kulli, V. R. Reverse Zagreb and reverse hyper-Zagreb indices and their polynomials of rhombus silicate networks. Ann. Pure Appl. Math. 16(1), 47–51 (2018).10.22457/apam.v16n1a6
65. Ravi V Siddiqui MK Chidambaram N Desikan K On topological descriptors and curvilinear regression analysis of antiviral drugs used in COVID-19 treatment Polycyclic Aromat. Compd. 2022 42 10 6932 6945 10.1080/10406638.2021.1993941
Ravi, V., Siddiqui, M. K., Chidambaram, N. & Desikan, K. On topological descriptors and curvilinear regression analysis of antiviral drugs used in COVID-19 treatment. Polycyclic Aromat. Compd. 42(10), 6932–6945 (2022).10.1080/10406638.2021.1993941
66. Arockiaraj M Greeni AB Kalaam AA Linear versus cubic regression models for analyzing generalized reverse degree based topological indices of certain latest corona treatment drug molecules Int. J. Quant. Chem. 2023 123 16 e27136 10.1002/qua.27136
Arockiaraj, M., Greeni, A. B. & Kalaam, A. A. Linear versus cubic regression models for analyzing generalized reverse degree based topological indices of certain latest corona treatment drug molecules. Int. J. Quant. Chem. 123(16), e27136 (2023).10.1002/qua.27136
67. Khan AR Bhatti SA Imran M Tawfiq FM Cancan M Hussain S Computation of differential and integral operators using M-polynomials of gold crystal Heliyon 2024 10 2024 e34419 10.1016/j.heliyon.2024.e34419 39149031
Khan, A. R. et al. Computation of differential and integral operators using M-polynomials of gold crystal. Heliyon 10(2024), e34419 (2024).39149031 10.1016/j.heliyon.2024.e34419
68. Xavier DA Julietraja K Alsinai A Akhila S Prediction of properties of graphyne by bond-addictive M-polynomial Sci. Rep. 2024 14 1 1197 10.1038/s41598-024-51642-2 38216625
Xavier, D. A., Julietraja, K., Alsinai, A. & Akhila, S. Prediction of properties of graphyne by bond-addictive M-polynomial. Sci. Rep. 14(1), 1197 (2024).38216625 10.1038/s41598-024-51642-2
69. Sabljic A Quantitative structure-toxicity relationship of chlorinated compounds: A molecular connectivity investigation Bull. Environ. Contam. Toxicol. 1983 30 80 83 10.1007/BF01610102 6403090
Sabljic, A. Quantitative structure-toxicity relationship of chlorinated compounds: A molecular connectivity investigation. Bull. Environ. Contam. Toxicol. 30, 80–83 (1983).6403090 10.1007/BF01610102
70. Shi M Hu W Li M Zhang J Song X Sun W Ensemble regression based on polynomial regression-based decision tree and its application in the in-situ data of tunnel boring machine Mech. Syst. Signal Process. 2023 188 110022 10.1016/j.ymssp.2022.110022
Shi, M. et al. Ensemble regression based on polynomial regression-based decision tree and its application in the in-situ data of tunnel boring machine. Mech. Syst. Signal Process. 188, 110022 (2023).10.1016/j.ymssp.2022.110022
71. Salzberg, S. L. C4. 5: Programs for machine learning by j. ross quinlan (Morgan Kaufmann Publishers, inc., 1993, 1994).
72. Breiman L Statistical modeling: The two cultures (with comments and a rejoinder by the author) Stat. Sci. 2001 16 3 199 231 10.1214/ss/1009213726
Breiman, L. Statistical modeling: The two cultures (with comments and a rejoinder by the author). Stat. Sci. 16(3), 199–231 (2001).10.1214/ss/1009213726
73. Liu M Meng F Liang Y Generalized pose decoupled network for unsupervised 3d skeleton sequence-based action representation learning Cyborg Bionic Syst. 2022 2022 0002 10.34133/cbsystems.0002 37040281
Liu, M., Meng, F. & Liang, Y. Generalized pose decoupled network for unsupervised 3d skeleton sequence-based action representation learning. Cyborg Bionic Syst. 2022, 0002 (2022).37040281 10.34133/cbsystems.0002
74. Imran M Ahmad A Siddiqui MK On degree-based topological descriptors of graphyne and graphdiyne nanoribbons Eur. Phys. J. Plus 2022 137 12 1372 10.1140/epjp/s13360-022-03514-9
Imran, M., Ahmad, A. & Siddiqui, M. K. On degree-based topological descriptors of graphyne and graphdiyne nanoribbons. Eur. Phys. J. Plus 137(12), 1372 (2022).10.1140/epjp/s13360-022-03514-9
75. Rahul MP Clement J Junias JS Arockiaraj M Balasubramanian K Degree-based entropies of graphene, graphyne and graphdiyne using Shannon’s approach J. Mol. Graph 2022 1260 132797
Rahul, M. P., Clement, J., Junias, J. S., Arockiaraj, M. & Balasubramanian, K. Degree-based entropies of graphene, graphyne and graphdiyne using Shannon’s approach. J. Mol. Graph 1260, 132797 (2022).
76. Jolliffe IT Principal component analysis for special types of data 2002 New York Springer 338 372
Jolliffe, I. T. Principal component analysis for special types of data 338–372 (Springer, New York, 2002).
77. Gauss, Carl Friedrich. “Theoria motus corporum coelestum.” Werke (1809).
