The matching energy of random graphs

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 is an element of (0, 1), Gutman and Wagner (2012) proposed a conjecture that the expectation of the matching energy of G(n,p) is asymptotically equal to 8 root p/3 pi n(3/2). In this paper, using analytical tools, we confirm this conjecture by obtaining a stronger result that the matching energy of G(n,p) is asymptotically almost surely equal to 8 root p/3 pi n(3/2). (C) 2015 Elsevier B.V. All rights reserved.