Permanental bounds for the signless Laplacian matrix of bipartite graphs and unicyclic graphs

Let G be a graph and let Q(G) be its signless Laplacian matrix. When G is a unicyclic (respectively, bipartite) graph we obtain sharp upper and lower bounds for the permanent of Q(G) in terms of the order of G. Improved bounds are obtained in terms of the given girth of G. In each of these cases, we characterize the extremal graphs.