On the maximum Zagreb indices of bipartite graphs with given connectivity

TL;DR

This paper establishes sharp upper bounds for the first and second Zagreb indices of bipartite graphs with fixed order and connectivity parameters, advancing understanding of their extremal properties.

Contribution

It provides the first precise bounds for Zagreb indices of bipartite graphs with given connectivity, identifying extremal graphs.

Findings

Sharp upper bounds for $M_1(G)$ and $M_2(G)$ are derived.

Extremal bipartite graphs achieving these bounds are characterized.

Results apply to graphs with specified vertex and edge connectivity.

Abstract

The first Zagreb index $M_{1}$ of a graph is defined as the sum of the square of every vertex degree, and the second Zagreb index $M_{2}$ of a graph is defined as the sum of the product of vertex degrees of each pair of adjacent vertices. In this paper, we study the Zagreb indices of bipartite graphs of order $n$ with $\kappa(G)=k$ (resp. $\kappa'(G)=s$) and sharp upper bounds are obtained for $M_1(G)$ and $M_2(G)$ for $G\in \mathcal{V}^k_n$ (resp. $\mathcal{E}^s_n$), where $\mathcal{V}^k_n$ is the set of bipartite graphs of order $n$ with $\kappa(G)=k$, and $\mathcal{E}^s_n$ is the set of bipartite graphs of order $n$ with $\kappa'(G)=s$.