On bipartite graphs with the minimum number of spanning trees

Abstract

The collection of all (simple and connected) bipartite graphs with cyclomatic number ω is denoted by $B_{\omega }$. We use $K_{a;b}^c$ to denote the graph obtained from the complete bipartite graph $K_{a,b+1}$ by removing $a-c$ edges that are all connected to the same vertex of degree a, here $a,b$ and c are integers with $2\le c<a\le b$. The term $S\left(G\right)$ denotes the skeleton of the graph G, which is defined as the largest induced subgraph of G that contains no pendant vertices.

In this paper, we investigate the problem of characterizing the graphs within $B_{\omega }$ that possess the minimum number of spanning trees. We show that the skeleton of each graph with the minimum number of spanning trees in $B_{\omega }$ is either $K_{a,b}$, where a and b are positive integers with $2\le a\le b$ and $(a-1)(b-1)=\omega$, or $K_{a;b}^c$, where $a,b$ and c are positive integers satisfying $2\le c<a\le b$ and $c-1+(a-1)(b-1)=\omega$. In addition, we establish some structural properties by the method of analysis to further reduce those candidate graphs.