Permanental roots and the star degree of a graph

For undirected graphs, without loops or multiple edges, we define the star degree of a graph, and prove that it is equal to the multiplicity of the root 1 of per(xI ? B), where B = D + A. Considering bipartite graphs, we prove that per(xI ? B) = per(xI ? L), where L = D ? A, and consequently that the star degree of a bipartite graph can also be characterized by the multiplicity of the root 1 of per(xI ? L).