Bounds on the Matching Energy of Unicyclic Odd-Cycle Graphs

Let G be a simple graph with order n and mu(1), mu(2),..., mu(n) be the roots of its matching polynomial. The matching energy of G is defined to be the sum of the absolute values of (i = 1, 2,...,n), which was proposed by Gutman and Wagner. Referring to graphs with no even cycles as odd-cycle graphs, denote by O-n the class of odd-cycle graphs of order it, and On,,n, the class of graphs in O-n with m edges. Especially, we call the graphs in as unicyclic odd-cycle graphs. In this paper, we determine the graphs with the second through the fourth maximal matching energies in O-n,O-n when n is odd, and establish the graphs with the maximal matching energy in O-n,O-n, when n is even. It is interesting that the extremal graphs for matching energy are of the form P-n(l) for some values of l, which are related to the extremal graph (i.e., P-n(6)) having the maximal energy among unicyclic graphs.