The matching energy of random graphs

TL;DR

This paper proves a conjecture that the matching energy of a random graph $G_{n,p}$ converges to a specific value proportional to $n^{3/2}$ as the number of vertices grows, confirming a long-standing hypothesis.

Contribution

The paper provides a rigorous proof confirming the conjecture about the asymptotic behavior of the matching energy in random graphs.

Findings

Matching energy of $G_{n,p}$ converges to $rac{8\sqrt{p}}{3\pi} n^{3/2}$ almost surely.

The proof uses analysis methods to establish the convergence.

Confirms a conjecture by Gutman and Wagner.

Abstract

The matching energy of a graph was introduced by Gutman and Wagner, which is defined as the sum of the absolute values of the roots of the matching polynomial of the graph. For the random graph $G_{n,p}$ of order $n$ with fixed probability $p\in (0,1)$, Gutman and Wagner [I. Gutman, S. Wagner, The matching energy of a graph, Discrete Appl. Math. 160(2012), 2177--2187] proposed a conjecture that the matching energy of $G_{n,p}$ converges to $\frac{8\sqrt{p}}{3\pi}n^{\frac{3}{2}}$ almost surely. In this paper, using analysis method, we prove that the conjecture is true.