# Spanning trees in complete bipartite graphs and resistance distance in nearly complete bipartite graphs

## TL;DR

This paper derives formulas for counting spanning trees in bipartite graphs using electrical network theory, extending existing results to nearly complete bipartite graphs and calculating related resistance indices.

## Contribution

It introduces a new formula for spanning trees in nearly complete bipartite graphs using resistance distance, generalizing prior specific cases.

## Key findings

- Formula for spanning trees in bipartite graphs with matchings or trees
- Extension of resistance distance results to nearly complete bipartite graphs
- Calculation of Kirchhoff index for these graphs

## Abstract

Using the theory of electrical network, we first obtain a simple formula for the number of spanning trees of a complete bipartite graph containing a certain matching or a certain tree. Then we apply the effective resistance (i.e., resistance distance in graphs) to find a formula for the number of spanning trees in the nearly complete bipartite graph $G(m,n,p)=K_{m,n}-pK_2$ $(p\leq \min\{m,n\})$, which extends a recent result by Ye and Yan who obtained the effective resistances and the number of spanning trees in $G(n,n,p)$. As a corollary, we obtain the Kirchhoff index of $G(m,n,p)$ which extends a previous result by Shi and Chen.