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 Gn,pGn,p of order nn with fixed probability p∈(0,1)p∈(0,1), Gutman and Wagner (2012) proposed a conjecture that the expectation of the matching energy of Gn,pGn,p is asymptotically equal to 8p3πn32. In this paper, using analytical tools, we confirm this conjecture by obtaining a stronger result that the matching energy of Gn,pGn,p is asymptotically almost surely equal to 8p3πn32.