A generalization of Boesch’s theorem

Boesch’s theorem says that “Suppose that a connected graph G has two non-adjacent 2-degree vertices u 1 and u 2 . Then t ( G ) ≤ t ( G / { u 1 , u 2 } ) , where t ( G ) is the number of spanning trees of G .” In this paper, we generalize this theorem as follows: “Suppose that G is a connected graph of order at least 3, and that u 1 and u 2 are two vertices of degree m and n , respectively, in G . Then t ( G ) ≤ m n − m 0 2 m + n − 2 m 0 t ( G / { u 1 , u 2 } ) , where m 0 ( ≥ 0 ) is the number of multiple edges between u 1 and u 2 .”