The coherent rank of a graph with three eigenvalues

TL;DR

This paper characterizes graphs with three eigenvalues and specific coherent ranks, linking them to combinatorial designs, discovering a new biregular graph, and proposing an infinite family of such graphs via switching techniques.

Contribution

It provides a characterization of graphs with three eigenvalues and specific coherent ranks, introduces a new biregular graph, and proposes an infinite family of graphs with these properties.

Findings

Characterization of graphs with three eigenvalues and coherent ranks 8 and 9.

Discovery of a new biregular graph with three eigenvalues.

Proposal of an infinite family of biregular graphs with three eigenvalues.

Abstract

We characterise graphs that have three distinct eigenvalues and coherent ranks 8 and 9, linking the former to certain symmetric 2-designs and the latter to specific quasi-symmetric 2-designs. This characterisation leads to the discovery of a new biregular graph with three distinct eigenvalues. Additionally, we demonstrate that the coherent rank of a triregular graph with three distinct eigenvalues is at least 14. Finally, we introduce a conjecturally infinite family of biregular graphs with three distinct eigenvalues, obtained by switching the block graphs of orthogonal arrays.