The bipartite unicyclic graphs with the first ⌊n−34⌋ largest matching energies

The theory of matching energy of graphs since been proposed by Gutman and Wagner in 2012, has attracted more and more attention. It is denoted by Bn,m,Bn,m, the class of bipartite graphs with order n and size m  . In particular, Bn,nBn,n denotes the set of bipartite unicyclic graphs, which is an interesting class of graphs. In this paper, for odd n  , we characterize the bipartite unicyclic graphs with the first ⌊n−34⌋ largest matching energies. There is an interesting correspondence: we conclude that the graph with the second maximal matching energy in Bn,nBn,n for odd n   ≥ 11 is Pn6, which is the only graph attaining the maximum value of the energy among all the (bipartite) unicyclic graphs for n ≥ 16.