Some Upper Bounds on the First General Zagreb Index

Muhammad Kamran Jamil,Iqra Zaman,Ebenezer Bonyah,Aisha Javed,Ali Ahmad

doi:10.1155/2022/8131346

Abstract

The first general Zagreb index M γ G or zeroth-order general Randić index of a graph G is defined as M γ G = ∑ v ∈ V d v γ where γ is any nonzero real number, d v is the degree of the vertex v and γ = 2 gives the classical first Zagreb index. The researchers investigated some sharp upper and lower bounds on zeroth-order general Randić index (for γ < 0 ) in terms of connectivity, minimum degree, and independent number. In this paper, we put sharp upper bounds on the first general Zagreb index in terms of independent number, minimum degree, and connectivity for γ . Furthermore, extremal graphs are also investigated which attained the upper bounds.

Highlights

Let G be a connected, simple, and finite graph with vertex set V(G) and edge set E(G). e total number of elements in V(G) is the order of the graph, and the number of edges which are connected with vertex v is said to be the degree d(v) of the vertex v
For nonadjacent vertices v and w, a vw-vertex cut is a subset R⊆V(G)\{v, w} where v and w are from different components of G − R and the smallest cardinality set of vertices which separates v and w is minimal vertex cut
A subset H⊆V(G) of nonadjacent vertices is called an independent set, and the largest cardinality set among all independent sets of G is the independent number

Summary

Introduction

Let G be a connected, simple, and finite graph with vertex set V(G) and edge set E(G). e total number of elements in V(G) is the order of the graph, and the number of edges which are connected with vertex v is said to be the degree d(v) of the vertex v. For nonadjacent vertices v and w, a vw-vertex cut is a subset R⊆V(G)\{v, w} where v and w are from different components of G − R and the smallest cardinality set of vertices which separates v and w is minimal vertex cut. If a smallest set of vertices V0 exists in a connected graph G whose deletion makes it disconnect, |V0| is said to be the vertex connectivity or simple connectivity of G. A topological index is a number corresponding to a molecule obtained from the molecular structure of the molecule. Is old and useful topological index helped in obtaining properties of the structure of molecules such as branching, ZE-isomerism, complexity, heterosystems, π-electron energy, and many more [3, 4].

Graphs with Given Connectivity and Minimum Degree

Bipartite Graphs with Given Connectivity

Graphs with Given Connectivity and Independent Number