# Computation of Topological Indices of NEPS of Graphs

Authors: Muhammad Imran; Shehnaz Akhter; Muhammad Kamran Jamil
Journal: Complexity
 (2021)

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## Abstract

The inspection of the networks and graphs through structural properties is a broad research topic with developing significance. One of the methods in analyzing structural properties is obtaining quantitative measures that encode data of the whole network by a real quantity. A large quantity of graph-associated numerical invariants has been used to examine the whole structure of networks. In this analysis, degree-related topological indices have a significant place in nanotechnology and theoretical chemistry. Thereby, the computation of indices is one of the successful branches of research. The noncomplete extendedp-sum NEPS of graphs is a famous general graph product. In this paper, we investigated the exact formulas of general zeroth-order Randić, Randić, and the first multiplicative Zagreb indices for NEPS of graphs.


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## Body

## 1. Introduction

Letℋ be a graph and Vℋ and Eℋ be the vertex and edge sets of ℋ, respectively. Vℋ denotes the number of vertices in ℋ, which is known as the order of ℋ, and the number of elements in Eℋ is said to be the size of ℋ, presented by Eℋ. The degree of a vertex v, presented as dℋv, in ℋ is the cardinality of linked edges with v. Graphs considered in this article are simple, finite, connected, and undirected. Graph theoretical notations which are not interpreted here can be seen in [1].Topological invariants have developed a lot of applications in drug discovery, including physicochemical, modeling information of molecules, toxicology, biology, and chemistry. There are numerous graphical invariants, which are effective in chemistry and nanotechnology. Thereby, the calculation of topological invariants is one of the worthwhile domains of graphical research. Suppose thatT corresponds to the set of all simple graphs and a function ℱ:T⟶ℛ+ is described as a topological invariant if for any set of two isomorphic ℋ1 and ℋ2 graphs, we have ℱℋ1=ℱℋ2. Hundreds of topological indices based on degree or distances have been introduced, but few of them gain popularity due to their high predictive power for various properties such as boiling point, density, refractive index, and molecular weight. For elaborated discussions on these invariants, we refer the reader to [2–11].Wiener proposed the first distance-related topological invariant, named the Wiener index, and is specified as [12](1)Wℋ=∑u′,v′⊆Vℋdℋu′,v′,where dℋu′,v′ is the shortest distance among u′ and v′ in ℋ. The authors in [13,14] investigated the Wiener index of several graph operations.For a graphℋ, Bollobás and Erdos [15] introduced the general Randić index as(2)Rγℋ=∑u′v′∈Eℋdℋu′dℋv′γ,where γ∈ℛ. The general Randić index is one of the most eye-catching topological indexes in QSAR and QSPR studies [16]. There is massive research related to the mathematical aspects of the general Randić index [10, 17]. Then, R1/2ℋ is the classical Randić index [18]. Zeroth-order general Randić index for ℋ is a modification of the Randić index [19, 20]. In 2005, Li and Zheng [21] presented the zeroth-order general Randić index for graphs, which is specified as(3)Rγ0ℋ=∑v′∈Vℋdℋv′γ,where γ∈ℛ.The firstM1 and second M2 Zagreb indices are very famous topological descriptors with several implementations in QSPR/QSAR. The first Zagreb index can be obtained by putting γ=2 in (3), while the second Zagreb index can be achieved by replacing γ=1 in (2) [22]. The chemical and mathematical aspects of the zeroth-order general Randić index have been investigated in [23–27].Recently, the authors in [28] proposed the general multiplicative Zagreb indices for ℋ. The first and the second general multiplicative Zagreb indices are described:(4)∏1,γℋ=∏u′∈Vℋdℋu′γ,∏2,γℋ=∏u′∈Vℋdℋu′γdℋu′,where γ∈ℛ.These are the generalizations of the classic first and second multiplicative Zagreb indices [29]. Khalifeh et al. [30] investigated the formulas of the Zagreb indices of graph products such as Cartesian product, composition, join, and disjunction. In [31], the authors investigated the bounds for multiplicative Zagreb indices of graph operations including the join, corona product, Cartesian product, composition, and disjunction.For graphsℋ1,ℋ2,…,ℋn and a given set ℬ⊆0,1n, the noncomplete extended p-sum NEPS of graph ℋl, 1≤l≤n, with respect to ℬ has the vertex set and the Cartesian product of Vℋl, 1≤l≤n, i.e., if ℋ=NEPSℋ1,ℋ2,…,ℋn;ℬ is NEPS of ℋ1,ℋ2,…,ℋn, then Vℋ=Vℋ1×Vℋ2×⋯×Vℋn. For u=u1,…,un,v=v1,…,vn∈Vℋ, there is an edge u1,…,unv1,…,vn∈Eℋ if and only if there are some n-tuple b1,…,bn∈ℬ such that ul=vl for bl=0, ul and vl are distinct, and uivl∈Eℋl for bl=1. In particular, NEPSℋ1;1=ℋ1, and NEPSℋ1;ϕ=NEPSℋ1;0 is the graph without any edge on the vertices of ℋ1.TheNEPS operations generalize several known graph products, the Cartesian product of graphs can be acquired by taking ℬ=b1,…,bn, where bi is an n-dimensional vector in which 1 is placed at the ith coordinate and 0 is placed elsewhere, and the tensor product of graphs can be achieved for ℬ=1,…,1.In 2012, Stevanoić [32] investigated the Zagreb indices for NEPS of graphs and showed that inequalities in conjecture M1ℋ/V≤M2ℋ/E unchanged under NEPS of graphs. In [33], the authors determined that the family of gcd-graphs and the family of NEPS of complete graphs coincide. For a more detailed study of NEPS and related topics, we refer the reader to [34–41]. In this paper, we extended the results of Stevanoić [32] and investigated the formulas of well-known degree-related topological descriptors for the NEPS of graphs.


## 2. Discussion and Main Results

We start this section with two well-known inequalities which will be beneficial in the proof of the main results.Lemma 1.
Letg1,g2,…,gn be nonnegative quantities. Then,(5)g1+g2+⋯+gnn≥g1g2,…,gnn,and the equality holds if and only if all gk are equal, 1≤k≤n.Lemma 2.
Letg1,g2,…,gn be nonnegative quantities and t1,t2,…,tn be nonnegative weights. Set t=t1+t2+⋯+tn. If t>0, then we have(6)t1g1+t2g2+⋯+tngnt≥g1g1g2t2,…,gntnt,and the equality holds if and only if all gk with tk>0 are equal, 1≤k≤n.Lemma 3 (see [42]).
Letℋ=NEPSℋ1,…,ℋn;ℬ; then, the degree of a vertex u=u1,u2,…,un is given as(7)dℋu=∑β∈ℬ∏i=1ndℋiβiui.Next, we derive the expression of the zeroth-order general Randić index of theNEPS of graphs for γ∈ℤ+.Theorem 1.
Letℋ=NEPSℋ1,ℋ2,…,ℋn;ℬ, where for l=1,…,n, n≥2, ℋl=Vl,El. If γ∈ℤ+, then(8)Rγ0ℋ=∑β1,…,βγ∈ℬγ∏l=1nRβl1+⋯+βlγ0ℋl.Proof.
Letℋ be the NEPS graph of ℋ1,ℋ2,…,ℋn. Then, by applying Lemma 3 in (3), we acquire(9)Rγ0ℋ=∑u∈Vℋdℋuγ=∑u1,…,un∈V1×⋯×Vn∑β∈ℬ∏l=1ndℋlβlulγ=∑u1,…,un∈V1×⋯×Vn∑β1∈ℬ∏l=1ndℋlβl1ul×∑β2∈ℬ∏l=1ndℋlβl2ul×…×∑βγ∈ℬ∏l=1ndℋlβlγul=∑u1,…,un∈V1×⋯×Vn∑β1,…,βγ∈ℬ×⋯×ℬ∏l=1ndℋlβl1+⋯+βlγul=∑β1,…,βγ∈ℬγ∑u1,…,un∈V1×⋯×Vn∏l=1ndℋlβl1+⋯+βlγul=∑β1,…,βγ∈ℬγ∏l=1n∑ul∈Vldℋlβl1+⋯+βlγul=∑β1,…,βγ∈ℬγ∏l=1nRβl1+⋯+βlγ0ul,which is the required result.
Forγ=2, the zeroth-order general Randić index gives the famous first Zagreb index, and we have the following corollary.Corollary 1 (see [32]).
Letℋ=NEPSℋ1,ℋ2,…,ℋn;ℬ, where ℋl=Vl,El for l=1,…,n, n≥2. Then, the first Zagreb index is(10)R20ℋ=∑β1,β2∈ℬ2∏l=1nRβl1+βl20ℋl.Corollary 2.
Letℋ=ℋ1⊗⋯⊗ℋn be the tensor product of graphs, where ℋl=Vl,El with l=1,…,n, n≥2, and γ∈ℤ+. Then,(11)Rγ0ℋ=∏l=1nRγ0ℋl.We proceed further to describe an upper bound of the zeroth-order general Randić index ofNEPS graphs for γ∈ℤ−.Theorem 2.
Letℋ=NEPSℋ1,ℋ2,…,ℋn;ℬ, where ℋl=Vl,El with l=1,…,n, n≥2, and γ∈ℤ−. Then,(12)Rγ0ℋ≥V2∑β1,…,βγ∈ℬγ∏l=1nRβl1+⋯+βlγℋl,and the equality holds if and only if ℋl’s, 1≤l≤n, are regular graphs.Proof.
Forγ∈ℤ−, using Lemma 3 in (3), we have(13)Rγ0ℋ=∑u∈Vℋdℋuγ=∑u1,…,un∈V1×⋯×Vn∑β∈ℬ∏l=1ndℋlβlulγ=∑u1,…,un∈V1×⋯×Vn1∑β1∈ℬ∏l=1ndℋlβl1ul×1∑β1∈ℬ∏l=1ndℋlβl2ul×⋯×1∑β1∈ℬ∏l=1ndℋlβlγul=∑u1,…,un∈V1×⋯×Vn1∑β1,…,βγ∈ℬ×⋯×ℬ∏l=1ndℋlβl1+⋯+βlγul≥V2∑u1,…,un∈V1×⋯×Vn∑β1,…,βγ∈ℬ×⋯×ℬ∏l=1ndℋlβl1+⋯+βlγul=V2∑β1,…,βγ∈ℬγ∑u1,u2,…,un∈V1×⋯×Vn∏l=1ndℋlβl1+⋯+βlγul=V2∑β1,…,βγ∈ℬγ∏l=1n∑ul∈Vldℋlβl1+⋯+βlγul=V2∑β1,…,βγ∈ℬγ∏l=1nRβl1+⋯+βlγ0ℋl,hence, the required result.Corollary 3.
Letℋ=ℋ1⊗⋯⊗ℋn be the tensor product of n graphs, where ℋl=Vl,El for l=1,2,…,n, n≥2, and γ∈ℤ−. Then,(14)Rγ0ℋ≥∏l=1nVl2∏l=1nRγ0ℋl,and the equality holds if and only if ℋl’s, 1≤l≤n, are regular graphs.In the upcoming theorem, we derive the result ofNEPS graphs with respect to the general Randić index.Theorem 3.
Letℋ=NEPSℋ1,ℋ2,…,ℋn;ℬ, where ℋl=Vl,El for l=1,…,n, n≥2. For γ∈ℤ+, the Randić index of ℋ is given as(15)Rγℋ=12∑t,β1,…,βγ,γ1,…,γγ∈ℬ2γ+1∏l=1n∑ul∈Vl,vl∈Vlul=vlfortl=0ulvl∈Elfortl=1dℋlAuldℋlBvl,where A=βl1+⋯+βlγ, B=γl1+⋯+γlγ, and 0≤A,B≤γ.Proof.
Letℋ=NEPSℋ1,ℋ2,…,ℋn;ℬ; then, the degree of each vertex is provided by Lemma 3. Thus, the definition of the general Randić index can be described as(16)Rγℋ=∑u,v∈Eℋdℋudℋvγ=12∑t∈ℬ∑u1,…,un∈V1×⋯×Vnv1,…,vn∈V1×⋯×Vnul=vlfortl=0uivl∈Elfortl=1∑β∈ℬ∏l=1ndℋlβlul∑γ∈ℬ∏l=1ndℋlγlvlγ=12∑t∈ℬ∑u1,…,un∈V1×⋯×Vnv1,…,vn∈V1×⋯×Vnul=vlfortl=0uivl∈Elfortl=1∑β1∈ℬ∏l=1ndℋlβl1ul∑γ1∈ℬ∏l=1ndℋlγl1vl×∑β2∈ℬ∏l=1ndℋlβl2ul∑γ2∈ℬ∏l=1ndℋlγl2vl×⋯×∑βγ∈ℬ∏l=1ndℋlβlγul∑γγ∈ℬ∏l=1ndℋlγlγvl=12∑t,β1,…,βγ,γ1,…,γγ∈ℬ2γ+1∑u1,…,un∈V1×⋯×Vnv1,…,vn∈V1×⋯×Vnul=vlfortl=0uivl∈Elfortl=1∏l=1ndℋlβl1+⋯+βlγuldℋlγl1+⋯+γlγvl=12∑t,β1,…,βγ,γ1,…,γγ∈ℬ2γ+1∏l=1n∑ul∈Vl,vl∈Vlul=vlfortl=0uivl∈Elfortl=1dℋlβl1+⋯+βlγuldℋlγl1+⋯+γlγvl=12∑t,β1,…,βγ,γ1,…,γγ∈ℬ2γ+1∏l=1n∑ul∈Vl,vl∈Vlul=vlfortl=0uivl∈Elfortl=1dℋlAuldℋlBvl,where A=βl1+⋯+βlγ, B=γl1+⋯+γlγ, and 0≤A,B≤γ.
IfA=B=K for some K∈ℤ+, then from the above result, we obtain the following:(17)Rγℋ=12∑t,β1,…,βγ,γ1,…,γγ∈ℬ2γ+1∏l=1n2RKℋl.
In the upcoming result, the general multiplicative Zagreb index ofNEPS of graphs ℋ1,ℋ2,…,ℋn is discussed.Theorem 4.
Letℋ=NEPSℋ1,ℋ2,…,ℋn;ℬ, where ℋl=Vl,El for l=1,…,n and n≥2. If γ∈ℤ+, then(18)∏1,γℋ≤∑β1,…,βγ∈ℬ×⋯×ℬ∏l=1nRβl1,…,βlγ0ℋlV|V,and the equality holds if and only if ℋls, 1≤l≤n, are regular graphs.Proof.
Using Lemma3, the definition of the general multiplicative Zagreb index for ℋ can be written as(19)∏1,γℋ=∏u∈Vℋdℋuγ=∏u1,…,un∈V1×⋯×Vn∑β∈ℬ∏l=1ndℋlβlulγ≤∑u1,…,un∈V1×⋯×Vn∑β∈ℬ∏l=1ndℋlβlulγVV=∑u1,…,un∈V1×…×Vn∑β1∈ℬ∏l=1ndℋlβl1ul×∑β2∈ℬ∏l=1ndℋlβl2ul×…×∑βγ∈ℬ∏l=1ndℋlβlγulVV=∑u1,…,un∈V1×⋯×Vn∑β1,…,βγ∈ℬ×⋯×ℬ∏l=1ndℋlβl1+⋯+βlγulVV=∑β1,…,βγ∈ℬ×⋯×ℬ∑u1,…,un∈V1×⋯×Vn∏l=1ndℋlβl1+⋯+βlγulVV=∑β1,…,βγ∈ℬ×⋯×ℬ∑ul∈Vl∏l=1ndℋlβl1+⋯+βlγulVV=∑β1,…,βγ∈ℬ×⋯×ℬ∏l=1nRβl1,…,βlγ0ℋlVV.
This completes the proof.


## 3. Conclusion

The investigation of graphs with the help of graph invariants is a very successful method, which performs an appreciable part in predicting the physicochemical characterizations of the given graphs. Thereby, the derivations of topological invariants are an attractive, appealing, and interesting line of research in the field of graph theory. The noncomplete extendedp-sum NEPS of graphs is a notable generalization of the graph product. In this paper, we have provided the results of the general zeroth-order Randić, Randić, and first multiplicative Zagreb indices for NEPS of graphs. It is more interesting to compute the eccentricity-based indices, PI, and Szeged and Mostar indices of NEPS of graphs.



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