Extremal matching energy of complements of trees

TL;DR

This paper characterizes trees based on the matching energy of their complements, identifying those with extremal values and exploring properties related to perfect matchings and edge-independence numbers.

Contribution

It provides a complete characterization of trees with extremal matching energy in their complements, including those with perfect matchings and specific independence numbers.

Findings

Identified trees with maximal, second-maximal, and minimal matching energy in their complements.

Determined trees with perfect matchings whose complements have second-maximal matching energy.

Showed trees with certain edge-independence numbers have minimum matching energy in their complements.

Abstract

The matching energy is defined as the sum of the absolute values of the zeros of the matching polynomial of a graph, which is proposed first by Gutman and Wagner [The matching energy of a graph, Discrete Appl. Math. 160 (2012) 2177--2187]. And they gave some properties and asymptotic results of the matching energy. In this paper, we characterize the trees with $n$ vertices whose complements have the maximal, second-maximal and minimal matching energy. Further, we determine the trees with a perfect matching whose complements have the second-maximal matching energy. In particular, show that the trees with edge-independence number number $p$ whose complements have the minimum matching energy for $p=1,2,\ldots, \lfloor\frac{n}{2}\rfloor$.