Multiplicity of integer roots of polynomials of graphs Original Research Article

Let G be a graph with minimum vertex degree p greater-or-equal, slanted 1. Let B = D + A, where D is the diagonal matrix of vertex degrees and A is the adjacency matrix of G. The multiplicity of the integer root p of per(xI − B) is characterized. For bipartite graphs, this characterization extends to per(xI − L), where L = D − A is the Laplacian matrix of G. For graphs with unrestricted vertex degrees, bounds are obtained on the multiplicities of integer roots of the permanental and characteristic polynomials of both L and B