# Topological Properties of Degree-Based Invariants via M-Polynomial Approach

Authors: Samirah Alsulami; Sabir Hussain; Farkhanda Afzal; Mohammad Reza Farahani; Deeba Afzal
Journal: Journal of Mathematics
 (2022)

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## Abstract

Chemical graph theory provides a link between molecular properties and a molecular graph. The M-polynomial is emerging as an efficient tool to recover the degree-based topological indices in chemical graph theory. In this work, we give the closed formulas of redefined first and second Zagreb indices, modified first Zagreb index, nano-Zagreb index, second hyper-Zagreb index, Randić index, reciprocal Randić index, first Gourava index, and product connectivity Gourava index via M-polynomial. We also present the M-polynomial of silicate network and then closed formulas of topological indices are applied on the silicate network.


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## Body

## 1. Introduction

Chemical graph theory (CGT) provides a connection between chemical structure and discrete mathematics. The ordered pair between the vertex set and edge set is considered a graph. In CGT, a graph is the replacement of molecular structure in which vertices are replaced by atoms and edges are considered as bonds. A topological index or invariant is an important tool of CGT that can guess the properties of the chemical species [1]. A mapping I:ϒ⟶ℝ is called a topological invariant if ∀G,H∈ϒ, GandH are isomorphic to each other if and only if IG=IH, where ϒ is the collection of molecular structure. Topological indices are a very useful application in structure-activity and structure-property modeling [2]. These are the best predictor to determine the physical, chemical, and biological behavior of the chemical substances [3].Without performing any testing, topological indices analyze many physicochemical properties of chemical compounds. Topological indices analyse the physical characteristics such as molar volume, melting point, surface tension, boiling point, molar refraction, and heats of vaporization.Topological indices also represent the biological behavior of compounds such as pH regulation, stimulation of cell growth, lipophilicity, toxicity, and nutrition. Hence, these topological indices may be helpful to know the chemical and physical characteristics and biological behaviors.The first topological index is the Wiener index which was initiated by Wiener in 1947 [4]. After that, thousands of indices have been designed till now [5]. A degree-dependent topological index for the graph G is defined as(1)IG=∑e=xy∈EGfdx,dy.In the chemical graph, by counting the same end-degree edges, then equation (1) can be rewritten as(2)IG=∑j≤kmjkfj,k,where dx,dy=j,k and the total number of edges xy is denoted by mjk.In 2013, Ranjini et al. introduced the new version of Zagreb indices known as redefined second and third Zagreb indices defined as [6](3)ReZG1=∑xy∈EGdx+dydxdy,ReZG2=∑xy∈EGdxdydx+dy.In 2003, the theory of Zagreb indices was reviewed in [7] written by Nikolić et al. and the new index was named as the modified first Zagreb index defined as(4)Mm1G=∑xy∈EG1dx+dy.In 2019, Jahanbani and Shooshtary [8] presented the nano-Zagreb index stated as:(5)NZG=∑xy∈EGdx2−dy2.Alameri [9] proposed the second hyper Zagreb index in 2021 and defined as(6)HM2G=∑xy∈EGdx2dy2.A bond-additive topological index was proposed by Randic in 1975 [10]. This index is a frequently examined index among all previously topological descriptors:(7)R−1/2G=∑xy∈EG1dx⋅dy.In 2014, Gutman et al. [11] presented the reciprocal Randić index defined as(8)RRG=∑xy∈EGdx⋅dy.In 2014, Kulli [12] presented the first Gourava index defined as(9)G1OG=∑xy∈EGdx+dy+dxdy.In 2017, Kulli [13] presented the product connectivity Gourava index and defined as(10)PGOG=∑xy∈EG1dx+dydxdy.By using different tools, we convert the molecular graph into some algebraic polynomial. Using these polynomials, the structural properties are determined easily [14]. These topological indices are either calculated directly by their formula or by using the graph polynomials such as M-polynomial. Deutsch and Klawzar defined the M-polynomial in 2015 and gave the nine closed formulas of the topological indices via M-polynomial [15]. After that, a lot of computation on different graphs was done in this area [16, 17]. In 2020, Afzal et al. proposed nine more closed formulas of topological indices by using the M-polynomial of the graph [18], and Hussain provided the relationship between the direct form and closed form via M-polynomial in 2021 [19]. In [20] Rajpoot and Selvaganesh presented another five closed forms of topological indices via M-polynomial. Shin et al. presented in 2021 a new set of nine topological indices calculated by the M-polynomial of the graph G [21]. In Table 1, we introduced nine more indices of degree-dependent topological indices via M-polynomial.Table 1
Topological indices derived fromMGu,v.
ReZG1G=SuSvDu+DvMGu,v|u=v=1ReZG2G=SuJDuDvMGu,v|u=1Mm1G=SuJMGu,v|u=1NZG=Dv2−Du2MGu,v|u=v=1HM2G=Du2Dv2MGu,v|u=v=1R−1/2G=Su1/2Sv1/2MGu,v|u=v=1RRG=Du1/2Dv1/2MGu,v|u=v=1G1OG=Du+Dv+DuDvMGu,v|u=v=1PGOG=Su1/2JSu1/2Sv1/2MGu,v|u=1M-polynomial for the graphG is defined as(11)MGu,v=∑ψ≤j≤k≤Ψmjkujvk.Here,ψ=mindx|x∈VG and Ψ=maxdx|x∈VG. Some operators, which are used in Table 1, are defined as(12)DuMGu,v=u∂∂uMGu,v,Du1/2MGu,v=u∂∂uMGu,v⋅MGu,v,DvMGu,v=v∂∂vMGu,v,Dv1/2MGu,v=v∂∂vMGu,v⋅MGu,v,SuMGu,v=∫0uMGt,vtdt,Su1/2MGu,v=∫0uMGt,vtdt⋅MGu,v,JMGu,v=MGu,u,Sv1/2MGu,v=∫0uMGt,vtdt⋅MGu,v.


## 2. Main Results

In the present section, proofs of closed formulas mentioned in Table1 are provided.Theorem 1.
IfMGu,v represents the M-polynomial for the graph G, then the redefined first Zagreb index is also calculated as ReZG1G=SuSvDu+DvMGu,v|u=v=1.Proof.
By taking(13)SuSvDu+DvMGu,v=SuSvDu+Dv∑ψ≤j≤k≤Ψmjkujvk=SuSvDu∑ψ≤j≤k≤Ψmjkujvk+Dv∑ψ≤j≤k≤Ψmjkujvk=SuSv∑ψ≤j≤k≤Ψjmjkujvk+∑ψ≤j≤k≤Ψkmjkujvk=SuSv∑ψ≤j≤k≤Ψj+kmjkujvk=Su∑ψ≤j≤k≤Ψj+kkmjkujvk=∑ψ≤j≤k≤Ψj+kjkmjkujvk,nowSuSvDu+DvMGu,v|u=v=1=∑ψ≤j≤k≤Ψj+kjkmjk=∑xy∈EGdx+dydxdy.
Hence,ReZG1G=SuSvDu+DvMGu,v|u=v=1u=v=1.Theorem 2.
IfMGu,v represents the M-polynomial of the graph G, then the redefined second Zagreb index is given by ReZG2G=SuJDuDvMGu,v|u=1.Proof.
By taking(14)SuJDuDvMGu,v=SuJDuDv∑ψ≤j≤k≤Ψmjkujvk=SuJDu∑ψ≤j≤k≤Ψkmjkujvk=SuJ∑ψ≤j≤k≤Ψjkmjkujvk=Su∑ψ≤j≤k≤Ψjkmjkuj+k=∑ψ≤j≤k≤Ψjkj+kmjkuj+k,nowSuJDuDvMGu,v|u=1=∑ψ≤j≤k≤Ψjkj+kmjk=∑ψ≤j≤k≤Ψdxdydx+dy.
Hence,ReZG2G=SuJDuDvMGu,v|u=1.Theorem 3.
IfMGu,v represents the M-polynomial of the graph G, then the modified first Zagreb index is computed by Mm1G=SuJMGu,v|u=1.Proof.
By taking(15)SuJMGu,v=SuJ∑ψ≤j≤k≤Ψmjkujvk=Su∑ψ≤j≤k≤Ψmjkuj+k=∑ψ≤j≤k≤Ψ1j+kmjkuj+k,nowSuJMGu,v|u=1=∑ψ≤j≤k≤Ψ1j+kmjk=∑xy∈EG1dx+dy.
Hence,Mm1G=SuJMGu,v|u=1Theorem 4.
IfMGu,v represents the M-polynomial of the graph G, then the nano-Zagreb index is given by NZG=Dv2−Du2MGu,v|u=v=1.Proof.
By taking(16)Dv2−Du2MGu,v=Dv2−Du2=∑ψ≤j≤k≤Ψmjkujvk=Dv2∑ψ≤j≤k≤Ψmjkujvk−Du2∑ψ≤j≤k≤Ψmjkujvk=∑ψ≤j≤k≤Ψk2mjkujvk−∑ψ≤j≤k≤Ψj2mjkujvk=−∑ψ≤j≤k≤Ψk2−j2mjkujvk,nowDv2−Du2MGu,v|u=v=1=∑ψ≤j≤k≤Ψk2−j2mjk=∑xy∈EGdx2−dy2.
Hence,NZG=Dv2−Du2MGu,v|u=v=1.Theorem 5.
IfMGu,v represents the M-polynomial of the graph G, then the second hyper-Zagreb index is computed as HM2G=Du2Dv2MGu,v|u=v=1.Proof.
By taking(17)Du2Dv2MGu,v=Du2Dv2=∑ψ≤j≤k≤Ψmjkujvk=Du2∑ψ≤j≤k≤Ψk2mjkujvk=∑ψ≤j≤k≤Ψj2k2mjkujvk,nowDx2Dy2MGu,v|u=v=1=∑ψ≤j≤k≤Ψj2k2mjk=∑ψ≤j≤k≤Ψdx2dy2.
Hence,HM2G=Dx2Dy2MGu,v|u=v=1.Theorem 6.
IfMGu,v represents the M-polynomial of the graph G, then the Randić index is computed as R−1/2G=Su1/2Sv1/2MGu,v|u=v=1.Proof.
By taking(18)Su1/2Sv1/2MGu,v=Su1/2Sv1/2=∑ψ≤j≤k≤Ψmjkujvk=Su1/2∑ψ≤j≤k≤Ψ1kmjkujvk=∑ψ≤j≤k≤Ψ1kjmjkujvk,nowSu1/2Sv1/2MGu,v|u=v=1=∑ψ≤j≤k≤Ψ1kjmjk=∑xy∈EG1dxdy.
Hence,R−1/2G=Su1/2Sv1/2MGu,v|u=v=1.Theorem 7.
IfMGu,v represents the M-polynomial for the graph G, then the reciprocal Randić index is given by RRG=Du1/2Dv1/2MGu,v|u=v=1.Proof.
By taking(19)Du1/2Dv1/2MGu,v=Du1/2Dv1/2∑xy∈EGmjkujvk=Du1/2∑ψ≤j≤k≤Ψkmjkujvk=∑ψ≤j≤k≤Ψjkmjkujvk,nowDu1/2Dv1/2MGu,v|u=v=1=∑ψ≤j≤k≤Ψjkmjk=∑xy∈EGdxdy.
Hence,RRG=Du1/2Dv1/2MGu,v|u=v=1Theorem 8.
IfMGu,v represents the M-polynomial of the graph G, then the first Gourava index is computed as G1OG=Du+Dv+DuDvMGu,v|u=v=1.Proof.
By taking(20)Du+Dv+DuDvMGu,v=Du+Dv+DuDv∑ψ≤j≤k≤Ψmjkujvk=Du∑ψ≤j≤k≤Ψmjkujvk+Dv∑ψ≤j≤k≤Ψmjkujvk+DuDv∑ψ≤j≤k≤Ψmjkujvk=∑ψ≤j≤k≤Ψjmjkujvk+∑ψ≤j≤k≤Ψkmjkujvk+Du∑ψ≤j≤k≤Ψkmjkujvk=∑ψ≤j≤k≤Ψj+kmjkujvk+∑ψ≤j≤k≤Ψjkmjkujvk=∑ψ≤j≤k≤Ψj+k+jkmjkujvk,nowDu+Dv+DuDvMGu,v|u=v=1=∑ψ≤j≤k≤Ψj+k+jkmjk=∑ψ≤j≤k≤Ψdx+dy+dxdy.
Hence,G1OG=Du+Dv+DuDvMGu,v|u=v=1.Theorem 9.
IfMGu,v represents the M-polynomial for the graph G, then the product connective Gourava index is computed as PGOG=Su1/2JSu1/2Sv1/2MGu,v|u=1.Proof.
By taking(21)Su1/2JSu1/2Sv1/2MGu,v=Su1/2JSu1/2Sv1/2∑ψ≤j≤k≤Ψmjkujvk=Su1/2JSu1/2∑ψ≤j≤k≤Ψ1kmjkujvk=Su1/2J∑ψ≤j≤k≤Ψ1jkmjkujvk=Su1/2∑ψ≤j≤k≤Ψ1jkmjkuj+k=∑ψ≤j≤k≤Ψ1j+kjkmjkuj+k,nowSu1/2JSu1/2Sv1/2MGu,v|u=1=∑ψ≤j≤k≤Ψ1j+kjkmjk=∑xy∈EG1dx+dydxdy.
Hence,PGOG=Su1/2JSu1/2Sv1/2MGu,v|u=1.


## 3. Chemical Graph of Silicate Network

The chemical graph of the silicate network (SLn) is shown in Figure 1, where n is the total number of hexagons present between the center of the network and the boundary of SLn. In Figure 1, blue and red dots represent the vertices having degrees 3 and 6, respectively. The edges having end-degrees 3,3, 3,6, and 6,6 are represented with yellow, green, and brown lines, respectively. Table 2 shows the vertex partitions, and Table 3 represents the edge partitions of SLn. In this present work, we extract some topological indices via M-polynomial of SLn.Figure 1
Chemical graph of silicate network (SL2).Table 2
Vertex partition ofSLn.
dxNumber of vertices36nn+163n3n−1Total vertices3n5n+1Table 3
Edge partition ofSLn.
dx,dyNumber of edges3,36n3,66n3n+16,66n3n−2Total edges36n2


## 4. M-Polynomial and Topological Invariants of Silicate Network

Theorem 10.
IfSLn represent a silicate network, then M-polynomial of SLn is MSLnu,v=6nu3v3+6n3n+1u3v6+6n3n−2u6v6 [22].The surface plot of M-polynomial of silicate network is shown in Figure2.Figure 2
Visual representation of M-polynomial of silicate network (SLn) for n = 2.Theorem 11.
LetSLn represent the silicate network and MSLnu,v=6nu3v3+6n3n+1u3v6+6n3n−2u6v6. Then,(1)
ReZG1SLn=15n2−3n(2)
ReZG2SLn=90n2−15n(3)
Mm1SLn=7/2n2+2/3n(4)
NZSLn=486n2+162n(5)
HM2SLn=29160n2−13122n(6)
R−1/2SLn=31+2n2+2n(7)
RRSLn=542+2−183−2n(8)
G1OSLn=1350n2−324n(9)
PGOSLn=1/222+3n2+1/32−3+6nProof.
(1)
The redefined first Zagreb index:DuMGu,v=18nu3v3+18n3n+1u3v6+36n3n−2u6v6DvMGu,v=18nu3v3+36n3n+1u3v6+36n3n−2u6v6Du+DvMGu,v=36nu3v3+54n3n+1u3v6+72n3n−2u6v6SvDu+DvMGu,v=12nu3v3+18n3n+1u3v6+12n3n−2u6v6SuSvDu+DvMGu,v=4nu3v3+3n3n+1u3v6+2n3n−2u6v6ReZG1G=DuDvDu+DvMGu,v|u=v=1=15n2−3n(2)
The redefined second Zagreb index:DvMGu,v=18nu3v3+36n3n+1u3v6+36n3n−2u6v6DuDvMGu,v=54nu3v3+108n3n+1u3v6+216n3n−2u6v6JDuDvMGu,v=54nu6+108n3n+1u9+216n3n−2u12SuJDuDvMGu,v=9nu6+12n3n+1u9+18n3n−2u12ReZG2G=SuJDuDvMGu,v|u=1=90n2−15n(3)
The modified first Zagreb index:JMGu,v=6nu6+6n3n+1u9+6n3n−2u12SuJMGu,v=nu6+2/3n3n+1u9+1/2n3n−2u12M1G=SuJMGu,v|u=1=7/2n2+2/3n(4)
The nano-Zagreb index:Dv2MGu,v=54nu3v3+216n3n+1u3v6+216n3n−2u6v6Du2MGu,v=54nu3v3+54n3n+1u3v6+216n3n−2u6v6Dv2−Du2MGu,v=162n3n+1u3v6NZSLn=Dv2−Du2MGu,v|u=v=1=486n2+162n(5)
The second hyper-Zagreb index:Dv2MGu,v=54nu3v3+216n3n+1u3v6+216n3n−2u6v6Du2Dv2MGu,v=486nu3v3+1944n3n+1u3v6+7776n3n−2u6v6HbSLn=Du2Dv2MGu,v|u=v=1=29160n2−13122n(6)
The Randić index:Sv1/2MGu,v=23nu3v3+6n3n+1u3v6+6n3n−2u6v6Su1/2Sv1/2MGu,v=2nu3v3+2n3n+1u3v6+n3n−2u6v6R−1/2G=Su1/2Sv1/2MGu,v|u=v=1=31+2n2+2n(7)
The reciprocal Randić index:Dv1/2MGu,v=63nu3v3+66n3n+1u3v6+66n3n−2u6v6Du1/2Dv1/2MGu,v=18nu3v3+182n3n+1u3v6+36n3n−2u6v6RRSLn=Du1/2Dv1/2MGu,v|u=v=1=542+2−183−2n(8)
The first Gourava index:DuMGu,v=18nu3v3+18n3n+1u3v6+36n3n−2u6v6DvMGu,v=18nu3v3+36n3n+1u3v6+36n3n−2u6v6DuDvMGu,v=54nu3v3+108n3n+1u3v6+216n3n−2u6v6Du+Dv+DuDvMGu,v=90nu3v3+162n3n+1u3v6+288n3n−2u6v6G1OSLn=Du+Dv+DuDvMGu,v|u=v=1=1350n2−324n(9)
The product connective Gourava index:Sv1/2MGu,v=23nu3v3+6n3n+1u3v6+6n3n−2u6v6Su1/2Sv1/2MGu,v=2nu3v3+2n3n+1u3v6+n3n−2u6v6JSu1/2Sv1/2MGu,v=2nu6+2n3n+1u9+n3n−2u12Su1/2JSu1/2Sv1/2MGu,v=6/3nu6+2/3n3n+1u9+3/6n3n−2u12GPOSLn=Su1/2JSu1/2Sv1/2MGu,v|u=1=1/222+3n2+1/32−3+6n
The plot of topological indices ofSLn is shown in Figure 3. According to the figure, product connective Gourava index, modified first Zagreb index, and redefined first Zagreb index have low values as compared to others.Figure 3
Plot of topological indices of silicate network (SLn).


## 5. Conclusion

In the paper, we gave some new formulas to find the vertex degree-dependent topological indices via M-polynomial. These formulas were then applied to the silicate network. These indices have valuable information about the molecular structure. The proposed new technique gave vast application in QSAR/QSPR to analyze the chemical structure. The results obtained are also plotted.



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