On the maximum signless Laplacian spectral radius of bipartite graphs

Suppose that the vertex set of a graph G is V (G) = {vi, center dot center dot center dot, v(n)}. Then we denote by d(v(i)) the degree of v(i) in G. Let A(G) be the adjacent matrix of G and D(G) be the n x n diagonal matrix with its (i, i)-entry equal to d(v(i)). Then Q(A)(G) = D(G) + A(G) is the signless Laplacian matrix of G. The signless Laplacian spectral radius of G is the largest eigenvalues of Q(A)(G). In this paper we describe the unique graph with maximum signless Laplacian spectral radius among all connected bipartite graphs of order n with a given matching number and vertex connectivity, respectively.