The matching energy of graphs with given edge connectivity

TL;DR

This paper investigates the maximum matching energy among connected graphs with a fixed order and edge connectivity, identifying a specific graph structure that maximizes this property.

Contribution

It establishes that the graph $K_{n-1,1}^k$ uniquely maximizes the matching energy among all connected graphs with given order and edge connectivity.

Findings

$K_{n-1,1}^k$ has the maximum matching energy among such graphs.

The result characterizes the extremal graph for matching energy under edge connectivity constraints.

Provides a new extremal graph theory result related to matching energy.

Abstract

Let G be a simple graph of order $n$ and $\mu_1,\mu_2,\ldots,\mu_n$ the roots of its matching polynomial. The matching energy of $G$ is defined as the sum $\sum_{i=1}^n|\mu_i|$. Let $K_{n-1,1}^k$ be the graph obtained from $K_1\cup K_{n-1}$ by adding $k$ edges between $V(K_1)$ and $V(K_{n-1})$. In this paper, we show that $K_{n-1,1}^k$ has maximum matching energy among all connected graph with order $n$ and edge connectivity $k$.