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 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. Also we establish the following upper bound for the number of spanning trees of G on n, e and d ₁ only: t(G)≤ (2e-d₁-1/n-2)^n-2. 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 d ₁ only.