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: 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. [ABSTRACT FROM AUTHOR]