# Degree-Based Topological Properties of Molecular Polymeric Networks Composed by Sierpinski Networks

Authors: Abdul Rauf; Muhammad Ishtiaq; Hafiz Faraz Qaiser; Adnan Aslam; Kraidi Anoh Yannick
Journal: Mathematical Problems in Engineering
 (2021)

---

## Abstract

Sierpinski graphs are a widely observed family of fractal-type graphs relevant to topology, Hanoi Tower mathematics, computer engineering, and around. Chemical implementations of graph theory establish significant properties, such as chemical activity, physicochemical properties, thermodynamic properties, and pharmacological activities of a molecular graph. Specific graph descriptors alluded to as topological indices are helpful to predict these properties. These graph descriptors have played a key role in quantitative structure-property/structure-activity relationships (QSPR/QSAR) research. The objective of this article is to compute Randic index (R−1/2), Zagreb index M1, sum-connectivity index SCI, geometric-arithmetic index GA, and atom-bond connectivity ABC index based on ev-degree and ve-degree for the Sierpinski networks Sn,m.


---

## Body

## 1. Introduction

Graph theory is concerned with network of points connected by lines. The beginning of graph theory deals with recreational math problems, but now it has grown into significant research area. Recently, it acquires sufficient attention due to its use in computer science, circuit boards, interrelated systems, living systems, sociology, and so on. Chemical graph theory (CGT) plays a very vital role in the composition and arrangement of substances. Chemical compounds are usually supposed to be the chemical networks, consequently whose compound’s atoms represent vertices, and bonds represent edges of a graph. Scientists are very attentive to know about the topology of chemical networks through some mathematical parameters gained from the network graphs of chemical molecules. A large scale of topological indices has been examined in biochemistry through QSPR/QSAR analysis to analyze chemical networks topology related to pharmacy, medical engineering, and experimental science, which are important fields of CGT [1]. A quick study of QSPR/QSAR is one of the main reasons to apply graph theory in chemistry [2, 3].The required technique for the study of some graphical structures is to break down the structures into substructures containing significant properties and also some associated structures that are supposed to have similar properties. We need to know how the network-related substructures are similar to each other in these circumstances. For example, to make polymeric networks, Sierpinski graphs are used. A class of polymeric materials which are directly or indirectly linked together as polymeric chains are polymeric networks. In mathematics, fractal theory and other areas of science, the Sierpinski, and these types of different graphs are used. In the recent decade, fractal nature-based graphs such as Sierpinski and related graphs were studied with applications in topology and computer science. Nowadays, the modern communication systems like radio and mobile radiations as far as pattern-based log-periodic behaviour have been studied by fractal nature of Sierpinski networks. The family of Sierpinski graphs has some unique properties and was discussed in the literature [4, 5].Topological indices are numerical numbers associated with the networks that can be helpful to predict some of its properties. Topological indices are mostly used for QSPR and QSAR studies. QSAR/QSPR studies have a lot of use in biochemistry and modern chemistry. It is important to select an appropriate descriptor to get a good correlation. There are many descriptors that can predict the simple properties of a molecular structure and give an insight into the properties of the structure under consideration [6]. Hence, we need to employ such molecular descriptors that can help to characterize the molecular properties and influence its activities [4, 7–9]. Harold Wiener introduces the concept of path number of a chemical graph and later calls it Wiener index. After that many topological indices were introduced this theory became the central topic of research. The most studied topological indices among these are the degree-based having many applications in chemical graph theory.Letℋ be a simple connected graph with its edge set and vertex set denoted by E and V, respectively. The order of ℋ is the cardinality of set V. Similarly, the size of ℋ is the cardinality of set E. Let u1∈V, then Nu1 denotes the set containing neighbors of u1 and is called open neighborhood of u1. The closed neighborhood of a vertex u1 is denoted by Nu1 and defined as Nu1=u1∪Nu1. We denote the degree of a vertex u1 by Λu1 and count of elements in Nu1. For basic concepts related to graph theory, we refer the readers to the book by West [10].First degree-based topological index was proposed by Randic [5] in 1975 and named as “branching index.” It was later called as Randic connectivity indexR−1/2ℋ. It was found to be useful to determine the degree of branching of saturated hydrocarbons in their carbon-atom skeleton. This index was later generalized by Bollobás and Erdos [11] in 1988 and they called it general Randic index. It was observed by Randic that a good correlation exists between various physical/chemical properties of alkanes and the Randic index Rα. The mathematical formulas for Randić index and general Randić index are(1)R−1/2ℋ=∑u1v1∈Eℋ1Λu1Λv1,Rαℋ=∑u1v1∈EℋΛu1Λv1α.Gutman introduces the first and second Zagreb indices in [12]. These indices were first applied to the branching problem [13]. The mathematical formulas for Zagreb indices are(2)M1ℋ=∑u1v1∈EℋΛu1+Λv1,M2ℋ=∑u1v1∈EℋΛu1×Λv1.The sum-connectivity index, denoted bySCI, was proposed by Zhou et al. [14] and has mathematical formula as(3)SCIℋ=∑u1v1∈Eℋ1Λu1+Λv1.Zhong [15] introduced the harmonic index as follows:(4)Hℋ=∑u1v1∈Eℋ2Λu1+Λv1.Estrada et al. [16] introduced the atom-bond connectivity ABC index which is formulated as(5)ABCℋ=∑u1v1∈EℋΛu1+Λv1−2Λu1Λv1.This index gives a good stability model for cycloalkanes and linear and branched alkanes strain energy [16, 17].Geometric-arithmetic indexGA was put forward by Vukičević et al. in [18] and is defined as(6)GAℋ=∑u1v1∈Eℋ2Λu1Λv1Λu1+Λv1.Using octane isomers, it was proved thatGA index shows good correlation with different physical/chemical properties.Recently, two new degree definitions, namely, ve-degree and ev-degree, were proposed by Chellali et al. [19]. Their mathematical properties were discussed in more detail later by Horoldagva et al. [20]. We denote the ev-degree of an edge u1v1=e1∈Eℋ by Λeve1 and is defined as the cardinality of the set Nu1∪Nv1. Similarly, for any vertex v1, its ve-degree is denoted by Λvev1 and is equal to the total number of edges that are incident to the vertices in the set Nv1. The classical degree-based topological indices definitions were converted into ev- and ve-degree. The Zagreb Mev index and Randic Rev index based on ev-degree for any edge e1=u1v1∈Eℋ are defined as(7)Mevℋ=∑e1∈EℋΛeve12,Revℋ=∑e1∈EℋΛeve1−1/2.The first Zagreb alphaM1αve index of a vertex v1∈Vℋ based on ve-degree is defined as(8)M1αveℋ=∑v1∈VℋΛvev12.Similarly, the ve-degree type first Zagreb betaM1βve index, geometric-arithmetic GAve index, atom-bond connectivity ABCve index, second Zagreb M2ve index, sum-connectivity χve index, Randic Rve index, and harmonic Hve index for each edge u1v1∈Eℋ are defined as(9)M1βveℋ=∑u1v1∈EℋΛveu1+Λvev1,M2veℋ=∑u1v1∈EℋΛveu1×Λvev1,ABCveℋ=∑u1v1∈EℋΛveu1+Λvev1−2Λveu1×Λvev1,GAveℋ=∑u1v1∈Eℋ2Λveu1×Λvev1Λveu1+Λvev1,Hveℋ=∑u1v1∈Eℋ2Λveu1+Λvev1,χveℋ=∑u1v1∈EℋΛveu1+Λvev1−1/2,Rveℋ=∑u1v1∈EℋΛveu1×Λvev1−1/2.A lot of research is going on these topological indices. The results related to these topological indices can be found in [21–25]. The objective of this paper is to calculate exact values of topological indices for Sierpinski network Sn,m based on ve- and ev-degree.


## 2. Methodology

For the calculation and computation of topological indexes such asMev, Rev, M1αve, M1βve, ABCve, M2ve, GAve, Hve, χve, and Rve indexes for polymeric molecular network composed by Sierpinski networks Sn,m, we compute the ev- and ve-degree of each edge and vertex, respectively. Moreover, for mathematical estimation, calculations, and confirmation, MATLAB software was used. To plot these mathematical results, we used the software maple.


## 3. Main Results

This section deals with the computation ofve- and ev-degree-based topological indices for Sierpinski network Sn,m. Let m,n≥1, then the Sierpinski network denoted by Sn,m is a graph on vertex set VSn,m=1,2,…,mn. Any vertex u∈VSn,m has the representation u=u1,u2,…,un, where ul∈1,2,…,m and l∈1,2,…,n. Two vertices u,v∈VSn,m are connected by an edge if and only if there exist and p∈1,2,…,n such that(i)
uq=vq for q=1,2,…,p−1.(ii)
up=vp, and(iii)
uq=vp and up=vq, for p=q+1,…,n.Sierpinski networksSn,m definition comes from the study of Lipscomb’s space. For n=3 in the definition of Sn,m, we get the tower of Hanoi graph. The graphs of S2,5 and S3,5 are depicted in Figure 1. It is easy to compute that the number of vertices and number of edges of Sn,m are mn and mn+1−m/2, respectively.Figure 1
(a) The Sierpinski network S (2, 5). (b) The Sierpinski network S (3, 5).
(a)(b)Theorem 1.
Letℋ be a graph of Sierpinski network Sn,m, then(10)Mevℋ=127mn+2−4mn−7m3−2m2+6m,Revℋ=m2−mm+1+mn−m22m+mn+1−mn−2m2+2m2m+2.Proof.
LetEp,q denote the set of edges of ℋ with end vertices of degree p and q, respectively. Then, Eℋ can be partitioned into two sets Em−1,mℋ and Em,mℋ. The ev-degree of each edge in the partition sets is presented in Table 1.
From Table1, the ev-degree-based Randic and Zagreb indices can be computed as(11)Mevℋ=∑e1∈EℋΛeve12,Mevℋ=m+12Em,m−1+2m2Em,m+m+22Em,m=m+12m2−m+2m2mn−m2+m+22mn+1−mn−2m2+2m2=127mn+2−4mn−7m3−2m2+6m,Revℋ=∑e1∈EℋΛeve1−1/2,Revℋ=m+1−1/2Em,m−1+2m−1/2Em,m+m+2−1/2Em,m=m+1−1/2m2−m+2m−1/2mn−m2+m+2−1/2mn+1−mn−2m2+2m2=m2−mm+1+mn−m22m+mn+1−mn−2m2+2m2m+2.Table 1
ev-degree of each edge ofℋ.
Λu1,Λv1Λeve1FrequencyEm,m−1m + 1mm−1Em,m2mmn−m/2Em,mm + 2mn+1−mn−2m2+2m/2Theorem 2.
Letℋ be a graph of Sierpinski network Sn,m, then(12)M1αveℋ=14mn+4+6mn+3+5mn+2−12mn+1+4mn−8m4−16m3+32m2−12m.Proof.
Letv1∈Vℋ, then the degree of v1 is either m−1 or m. The ve-degree of a vertex is either m2+m−2/2 or m2+3m−4/2. These data are presented in Table 2. Using Table 2, the value of M1αveℋ can be calculated asTable 2
ve-degree of each vertex ofℋ.
Λu1Λveu1Frequencym−1m2+m−2/2mmm2+3m−4/2m2−mmm2+3m−2/2mn−m2(13)M1αveℋ=∑v1∈VℋΛvev12,M1αveℋ=m2+m−222m+m2+3m−422m2−m+m2+3m−422mn−m=14mn+4+6mn+3+5mn+2−12mn+1+4mn−8m4−16m3+32m2−12m.Theorem 3.
Letℋ be a molecular graph of Sierpinski network Sn,m, then(14)M1βveℋ=142mn+3+6mn+2−4mn+1−10m3+2m2+4m,M2veℋ=18mn+5+6mn+4+5mn+3−12mn+2+4mn+1−9m5−22m4+47m3−24m2+4m,ABCveℋ=2m2−mm2+2m−5m4+4m3−3m2−10m+8+m2+3m−5m4+6m3+3m2−18m+m3−3m2+2mm2+3m−6m4+6m3+m2−24m+16+14mn+1−m3−m2+mm2+3m−4m4+5m2+6m3−12m+4,GAveℋ=m2−mm4+4m3−3m2−10m+8m2+2m−3+m4+6m3+3m2−18mm2+3m−3+12m3−3m2+2mm4+6m3+m2−24m+16m2+3m−4+mn+1−m3−m2+mm4+5m2+6m3−12m+4m2+3m−2,Hveℋ=2mm+3+2m2−mm2+3m−3+m2−2mm+4+mmn−m2−m+1m2+3m−2,χveℋ=m2−m1m2+2m−3+1m2+3m−3+12m3−3m2+2mm2+3m−4+mn+1−m3−m2+mm2+3m−2,Rveℋ=2m2−m1m4+4m3−3m2−10m+8+1m4+6m3+3m2−18m+8+m3−3m2+2mm4+6m3+m2−24m+16+mn+1−m3−m2+mm4+6m3+5m2−12m+4.Proof.
Based on the degree of end vertices of each edge, we can partitionEℋ into two sets Em−1,m and Em,m. Now, depending on the ve-degree of end vertex of each edge, we can further partition the sets Em−1,m and Em,m into four sets. Table 3 shows the details of this partition.Table 3
ev-degree of each edge ofℋ.
EdgeΛveu1,Λvev1FrequencyEm−1,m∗m2+m−2/2,m2+3m−4/2m2−mEm,m∗m2+3m−4/2,m2+3m−2/2m2−mEm,m∗m2+3m−4/2,m2+3m−4/21/2m3−3m2+2mEm,m∗m2+3m−2/2,m2+3m−2/21/2mn+1−m3−m2+mUsing Table3, we can compute the indices as(15)M1βveℋ=∑u1v1∈EℋΛveu1+Λvev1,M1βveℋ=m2+m−22+m2+3m−42Em−1,m∗+m2+3m−42+m2+3m−22Em−1,m∗+m2+3m−42+m2+3m−42Em−1,m∗+m2+3m−22+m2+3m−22Em−1,m∗=m2+2m−3m2−m+m2+3m−3m2−m+12m2+3m−4m3−3m2+2m+12m2+3m−2mn+1−m3−m2+m=142mn+3+6mn+2−4mn+1−10m3+2m2+4m,M2veℋ=∑u1v1∈EℋΛveu1Λvev1,M2veℋ=m2+m−22m2+3m−42Em−1,m∗+m2+3m−42m2+3m−22Em−1,m∗+m2+3m−42m2+3m−42Em−1,m∗+m2+3m−22m2+3m−22Em−1,m∗=14m4+4m3−3m2−10m+8m2−m+14m2−mm4+6m3+3m2−18m+8+18m4+6m3+m2−24m+16m3−3m2+2m+18m4+6m3+5m2−12m+4mn+1−m3−m2+m=18mn+5+6mn+4+5mn+3−12mn+2+4mn+1−9m5−22m4+47m3−24m2+4m.


## 4. Numerical Discussions and Results

In this portion, we have represented the numerical outcomes of the above computed topological descriptors related to the ev-degrees and ve-degrees for molecular polymeric networks composed by Sierpinski network. We have computed numerical tables/results by using different values ofn and k(see Tables 4, 5, and 6). Moreover, we have drawn the graphical results (see Figures 2–7) to test and review the behaviour of topological descriptors computed above. It is clear from Tables 4, 5, and 6 and Figures 2–Figure 7 that, as there is increase in n and k, the computed topological descriptors are increasing.Table 4
Numerical table related to ev-degree-based indices.
n,k1,12,23,34,45,56,6Mev022702135962.6674e + 055.7846e + 06Rev01.654717.29201.782856.214282Table 5
Numerical table related to ve-degree index.
n,m1,12,23,34,45,56,67,7M1αve026152142612112656031536270952009905Table 6
Numerical table related to end vertices ve-degrees indices.
n,m1,12,23,34,45,56,6M1βve016588131642965807277820M2ve0202229850022.8157e + 069.4607e + 07ABCve00.996545.494639.799513.588978.6GAve1403774.741403778.721403786.681403798.631403814.551403834.46Hve0.31.12145.228639.564411.375383.6χve149171491714928149761509715342Rve01.14385.244539.577411.385383.6Figure 2
(a) The graphMevℋ. (b) The graph of Revℋ.
(a)(b)Figure 3
The graph ofM1αveℋ.Figure 4
(a) The graph ofM1βveℋ. (b) The graph of M2veℋ.
(a)(b)(16)ABCveℋ=∑u1v1∈EℋΛveu1+Λvev1−2Λveu1Λvev1ABCveℋ=2m2+2m−5m4+4m3−3m2−10m+8Em−1,m∗+2m2+3m−5m4+6m3+3m2−18mEm−1,m∗+2m2+3m−6m4+6m3+m2−24m+16Em−1,m∗+2m2+3m−4m4+5m2+6m3−12m+4Em−1,m∗=2m2−mm2+2m−5m4+4m3−3m2−10m+8+2m2−mm2+3m−5m4+6m3+3m2−18m+m3−3m2+2mm2+3m−6m4+6m3+m2−24m+16+14mn+1−m3−m2+mm2+3m−4m4+5m2+6m3−12m+4=2m2−mm2+2m−5m4+4m3−3m2−10m+8+m2+3m−5m4+6m3+3m2−18m+m3−3m2+2mm2+3m−6m4+6m3+m2−24m+16+14mn+1−m3−m2+mm2+3m−4m4+5m2+6m3−12m+4,GAveℋ=∑u1v1∈Eℋ2Λveu1×Λvev1Λveu1+Λvev1,GAveℋ=m4+4m3−3m2−10m+8m2+2m−3Em−1,m∗+m4+6m3+3m2−18mm2+3m−3Em−1,m∗+m4+6m3+m2−24m+162m2+3m−4Em−1,m∗+m4+6m3+5m2−12m+42m2+3m−2Em−1,m∗=m2−mm4+4m3−3m2−10m+8m2+2m−3+m2−mm4+6m3+3m2−18mm2+3m−3+m3−3m2+2mm4+6m3+m2−24m+162m2+3m−4+mn+1−m3−m2+mm4+6m3+5m2−12m+42m2+3m−2=m2−mm4+4m3−3m2−10m+8m2+2m−3+m4+6m3+3m2−18mm2+3m−3+m3−3m2+2mm4+6m3+m2−24m+162m2+3m−4+mn+1−m3−m2+mm4+6m3+5m2−12m+42m2+3m−2.Figure 5
(a) The graph ofABCveℋ. (b) The graph of GAveℋ.
(a)(b)(17)Hveℋ=∑u1v1∈Eℋ2Λveu1+Λvev1Hveℋ=2m2+2m−3Em−1,m∗+2m2+3m−3Em,m∗+1m2+3m−4Em,m∗+1m2+3m−2Em,m∗=2m2+2m−3m2−m+2m2+3m−3m2−m+12m2+3m−4m3−3m2+2m+1m2+3m−2mn+1−m3−m2+m=2mm+3+2mm−1m2+3m−3+mm−2m+4+mmn−m2−m+1m2+3m−2,χveℋ=∑u1v1∈EℋΛveu1+Λvev1−1/2=χveℋ=m2+2m−3−1/2Em−1,m∗+m2+3m−3−1/2Em−1,m∗+m2+3m−4−1/2Em−1,m∗+m2+3m−2−1/2Em−1,m∗=m2+2m−3−1/2m2−m+m2+3m−3−1/2m2−m+12m2+3m−4−1/2m3−3m2+2m+12m2+3m−2−1/2mn+1−m3−m2+m+12m3−3m2+2mm2+3m−4+mn+1−m3−m2+mm2+3m−2.Figure 6
(a) The graph ofHveℋ. (b) The graph of χveℋ.
(a)(b)(18)Rveℋ=∑u1v1∈EℋΛveu1Λvev1−1/2,Rveℋ=14m4+4m3−3m2−10m+8−1/2Em−1,m∗+14m4+6m3+3m2−18m+8−1/2Em−1,m∗+14m4+6m3+m2−24m+16−1/2Em−1,m∗+14m4+6m3+5m2−12m+4−1/2Em−1,m∗=14m4+4m3−3m2−10m+8−1/2m2−m+14m4+6m3+3m2−18m+8−1/2m2−m+1214m4+6m3+m2−24m+16−1/2m3−3m2+2m+1214m4+6m3+5m2−12m+4−1/2mn+1−m3−m2+m=2m2−mm4+4m3−3m2−10m+8+2m2−mm4+6m3+3m2−18m+8+m3−3m2+2mm4+6m3+m2−24m+16+mn+1−m3−m2+mm4+6m3+5m2−12m+4.Figure 7
The graph ofRveℋ.


## 5. Conclusion

To understand the basic topology, the analysis of chemical graphs and networks by topological descriptors is required. Such research has a broad variety of uses in the fields of bioinformatics, cheminformatics, biostatistics, and drug delivery, where a variety of daunting schemes are tackled by diagnostics based on different graph endeavours. Graphs are essential methods for estimating and forecasting the characteristics of chemical and biological compounds in the study of QSPR and QSAR.In this paper, we have presented the results for ev- and ve-degree-based topological indices for the polymeric molecular networks modeled by Sierpinski networks that can be utilized for QSPR and QSAR study.



---

*Source: 2066815-2021-10-08.xml*