Counterexamples of the Bhattacharya-Friedland-Peled conjecture

TL;DR

This paper presents counterexamples to the BFP conjecture, an analog of a spectral extremal graph conjecture for bipartite graphs, showing that the conjecture does not hold universally.

Contribution

The paper provides the first known counterexamples to the BFP conjecture, challenging its validity in the spectral graph theory context.

Findings

Counterexamples disprove the BFP conjecture

The conjecture holds only under certain conditions

Highlights limitations of spectral extremal graph conjectures

Abstract

The Brauldi-Hoffman conjecture, proved by Rowlinson in 1988, characterized the graph with maximal spectral radius among all simple graphs with prescribed number of edges. In 2008, Bhattacharya, Friedland, and Peled proposed an analog, which will be called the BFP conjecture in the following, of the Brauldi-Hoffman conjecture for the bipartite graphs with fixed numbers of edges in the graph and vertices in the bipartition. The BFP conjecture was proved to be correct if the number of edges is large enough by several authors. However, in this paper we provide some counterexamples of the BFP conjecture.