Undirected simple connected graphs with minimum number of spanning trees

We show that for positive integers n , m with n ( n − 1 ) / 2 ≥ m ≥ n − 1 , the graph L n , m having n vertices and m edges that consists of an ( n − k ) -clique and k − 1 vertices of degree 1 has the fewest spanning trees among all connected graphs on n vertices and m edges. This proves Boesch’s conjecture [F.T. Boesch, A. Satyanarayana, C.L. Suffel, Least reliable networks and reliability domination, IEEE Trans. Commun. 38 (1990) 2004–2009].