A Sharp Upper Bound for the Number of Spanning Trees of a Graph

摘要

Let G = (V,E) be a simple graph with n vertices, e edges and d 1 be the highest degree. Further let λ i , i = 1,2,...,n be the non-increasing eigenvalues of the Laplacian matrix of the graph G. In this paper, we obtain the following result: For connected graph G, λ2 = λ3 = ... = λ n-1 if and only if G is a complete graph or a star graph or a (d 1,d 1) complete bipartite graph.