The number of spanning forests of a graph Original Research Article

In this paper we study the number of spanning forests of a graph. Let image be a connected simple graph. (1) We give a lower bound for the number of spanning forests of image in terms of the edge connectivity of image. (2) We give an upper bound for the number of rooted spanning forests of image. (3) We describe the elementary symmetric functions of inverse positive Laplacian eigenvalues of a tree. (4) We determine all Laplacian integral graphs with prime number of spanning trees. (5) We give a simple proof of a theorem of K. Hashimoto on Ihara zeta function.