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

Let G = (V,E) be a simple graph with n vertices, e edges and d₁ 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 (d₁,d₁) complete bipartite graph. Also we establish the following upper bound for the number of spanning trees of G on n, e and d₁ only: The equality holds if and only if G is a star graph or a complete graph. Earlier bounds by Grimmett 5], Grone and Merris 6], Nosal 11], and Kelmans 2] were sharp for complete graphs only. Also our bound depends on n, e and d₁ only. This work was done while the author was doing postdoctoral research in LRI, Université Paris-XI, Orsay, France.