The number of spanning trees in Km,n-complements of bipartite graphs

Abstract

For a subgraph $G$ in the complete bipartite graph $K_{m,n}$, the $K_{m,n}$-complement of $G$ (namely $K_{m,n}-G$) is the graph obtained from $K_{m,n}$ by removing the edges of $G$. In this paper, by the matrix-tree theorem, we derive a general expression for the number of spanning trees of $K_{m,n}-G$ and establish explicit formulas for the number of spanning trees of the $K_{m,n}$-complements of various classes of bipartite graphs, which generalizes some known results.