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 mu(i)(i = 1, 2, ... , n), which was proposed by Gutman and Wagner. A maximum matching is one which covers as many vertices as possible. The number of edges in a maximum matching of a graph G is called the matching number of G and denoted by alpha' (G). Let B(n, beta) and UB(n, beta) be the set of connected bipartite graphs and connected bipartite unicyclic graphs with order n and matching number beta, respectively. In this paper, we characterize graphs with the first three largest matching energies in B(n, beta). Also we determine the extremal graph with minimal and the second minimal matching energy among graphs in B(n, beta), respectively. Furthermore, we determine the extremal graph from UB(n, beta) minimizing the matching energy.
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