Digital Object Identifier (DOI) 10.1007/s00373-007-0758-4 Graphs and Combinatorics (2007) 23:625–632 Graphs and Combinatorics © Springer 2007 A Sharp Upper Bound for the Number of Spanning Trees of a Graph Kinkar Ch. Das  Universit ´ e Paris-XI, Orsay, LRI, B ˆ atiment 490, 91405 Orsay Cedex, France. e-mail: kinkar@mailcity.com Abstract. 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, λ 2 = λ 3 = ... = 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 − d 1 − 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. Key words. Graph, spanning trees, Laplacian matrix. 1. Introduction Let G = (V,E) be a simple graph with the vertex set V ={v 1 ,v 2 ,... ,v n } and the cardinality of edge set e. Assume that the vertices are ordered such that d 1 ≥ d 2 ≥ ... ≥ d n , where d i is the degree of v i for i = 1, 2,... ,n. The number of span- ning trees of G is denoted by t (G). Let A(G) be the (0, 1)-adjacency matrix of G and D(G) be the diagonal matrix of vertex degrees. The Laplacian matrix of G is L(G) = D(G) − A(G). Clearly, L(G) is a real symmetric matrix. From this fact and Ger ˇ s gorin’s theorem, it follows that its eigenvalues are non-negative real numbers. Moreover since its rows sum is equal to 0, 0 is the smallest eigenvalue of L(G). It is known that the multiplicity of 0 as the eigenvalue of L(G) is equal to the number of connected components of G. So a graph G is connected if and only if the second smallest Laplacian eigenvalue is strictly greater than 0. Throughout this paper let λ 1 ≥ λ 2 ≥ ... ≥ λ n−1 ≥ λ n = 0 be the eigenvalues of L(G). When more than  This work was done while the author was doing postdoctoral research in LRI, Universit ´ e Paris-XI, Orsay, France.