Comparison of graphs by their number of spanning trees

Abstract

The problem of comparison of graphs with the same number of vertices and edges by their number of spanning trees is considered. A set of operations on graphs which increase the number of their spanning trees is given. In particular, the following assertions are proved: (1) A disconnected graph is “better” (it destroys less trees if removed from the complete graph) than any connected separable graph whose blocks are components of the given graph. (2) The replacement, in a separable graph, of a block B with m edges which hangs at the unique vertex x, by the star with m edges fixed at its center in x, “worsens” the graph. (3) A chain (star) composed of identical and symmetric two-terminal networks is “better” (“worse”) than any other tree composed from the same two-terminal networks. (4) A chain (a circle) is “better” than any connected (respectively 2-connected) graph with the same number of edges. The article is concluded by a description of some operations on graphs which permit the extension of the results to a wider class of graphs.