Upper Bounds for the Modified Second Multiplicative Zagreb Index of Graph Operations

B Basavanagoud ,Shreekant Patil

doi:10.18052/www.scipress.com/bmsa.17.10

B Basavanagoud , Shreekant Patil

Open Access

https://doi.org/10.18052/www.scipress.com/bmsa.17.10

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Abstract

The modified second multiplicative Zagreb index of a connected graph G, denoted by $\prod_{2}^{*}(G)$, is defined as $\prod_{2}^{*}(G)=\prod \limits_{uv\in E(G)}[d_{G}(u)+d_{G}(v)]^{[d_{G}(u)+d_{G}(v)]}$ where $d_{G}(z)$ is the degree of a vertex z in G. In this paper, we present some upper bounds for the modified second multiplicative Zagreb index of graph operations such as union, join, Cartesian product, composition and corona product of graphs are derived.The modified second multiplicative Zagreb index of aconnected graph , denoted by , is defined as where is the degree of avertex in . In this paper, we present some upper bounds for themodified second multiplicative Zagreb index of graph operations such as union,join, Cartesian product, composition and corona product of graphs are derived.

Highlights

Throughout this paper, we consider only simple connected graphs
We present some upper bounds for the modified second multiplicative Zagreb index of graph operations such as union, join, Cartesian product, composition and corona product of graphs are derived
Sayadi [13] introduced a new version of Zagreb index named hyper-Zagreb index which is defined for a graph G as HM(G) = ∑uv∈E(G) ‍(dG(u) + dG(v)

Summary

Introduction

Let G be such a graph on n vertices and m edges. If u and v are two adjacent vertices of G, the edge connecting them will be denoted by uv. The degree of a vertex w ∈ V(G) is the number of vertices adjacent to w and is denoted by dG(w). The first and second Zagreb indices, respectively, defined. M. Sayadi [13] introduced a new version of Zagreb index named hyper-Zagreb index which is defined for a graph G as HM(G) = ∑uv∈E(G) ‍(dG(u) + dG(v)). Basavanagoud et al [5] introduced a multiplicative version of index named modified second multiplicative Zagreb index and is defined as. Xnwn holds with equality if and only if all the xk with wk > 0 are equal

Main results

Cartesian product

Composition