Minimum number of distinct eigenvalues of distance-regular and signed Johnson graphs

TL;DR

This paper investigates the minimum number of distinct eigenvalues in matrices associated with various graph classes, revealing that Johnson graphs have signed variants with only two eigenvalues and exploring multiple applications.

Contribution

It proves Johnson graphs have signed variants with exactly two eigenvalues and extends results to association schemes, Hamming graphs, and related structures.

Findings

Johnson graphs have signed variants with two eigenvalues

Lower bounds on eigenvalues depend on graph cycles

Applications to coding theory and combinatorial designs

Abstract

We study the minimum number of distinct eigenvalues over a collection of matrices associated with a graph. Lower bounds are derived based on the existence or non-existence of certain cycle(s) in a graph. A key result proves that every Johnson graph has a signed variant with exactly two distinct eigenvalues. We also explore applications to weighing matrices, linear ternary codes, tight frames, and compute the minimum rank of Johnson graphs. Further results involve the minimum number of distinct eigenvalues for graphs in association schemes, distance-regular graphs, and Hamming graphs. We also draw some connections with simplicial complexes and higher-order Laplacians.