Maximizing spectral radius and number of spanning trees in bipartite graphs

TL;DR

This paper investigates extremal bipartite graphs for spectral radius and spanning trees, providing new proofs and supporting conjectures that Ferrers graphs are optimal under certain degree constraints.

Contribution

It offers a new proof for the spanning trees formula in Ferrers graphs and discusses conjectures about their optimality for spectral radius and spanning trees.

Findings

Confirmed the formula for spanning trees in Ferrers graphs

Supported the conjecture that Ferrers graphs maximize spectral radius

Provided conditions under which edge removal does not affect resistance distance

Abstract

The problems of maximizing the spectral radius and the number of spanning trees in a class of bipartite graphs with certain degree constraints are considered. In both the problems, the optimal graph is conjectured to be a Ferrers graph. Known results towards the resolution of the conjectures are described. We give yet another proof of a formula due to Ehrenborg and van Willigenburg for the number of spanning trees in a Ferrers graph. The main tool is a result which gives several necessary and sufficient conditions under which the removal of an edge in a graph does not affect the resistance distance between the end-vertices of another edge.