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 λi, 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 = ... =  λn-1 if and only if G is a complete graph or a star graph or a (d1,d1) complete bipartite graph.