The Extremal Matching Energy of Graphs

Abstract

Let \(G\) be a simple graph of order \(n\) with \(\mu_1, \mu_2, \ldots, \mu_n\) as the roots of its matching polynomial. Recently, Gutman and Wagner defined the matching energy as \(\sum_{i=1}^{n} |\mu_i|\). In this paper, we first show that the Turán graph \(T_{r,n}\) is the \(r\)-partite graph of order \(n\) with maximum matching energy. Furthermore, we characterize the connected graphs (and bipartite graphs) of order \(n\) having minimum matching energy with \(m\) edges, where \(n+2 \leq m \leq 2n-4\) (and \(n \leq m\leq 2n-5\)).