Orthogonal symmetric matrices and joins of graphs

TL;DR

The paper investigates conditions under which the join of two graphs has a symmetric orthogonal matrix pattern with only two distinct eigenvalues, providing necessary and sufficient criteria and characterizing specific cases.

Contribution

It introduces a compatibility notion for multiplicity matrices and establishes new conditions for the eigenvalue structure of graph joins.

Findings

Necessary condition for the join to have a symmetric orthogonal matrix with two eigenvalues.

Sufficient condition under additional hypotheses.

Characterization of the eigenvalue count for joins of unions of complete graphs.

Abstract

We introduce a notion of compatibility for multiplicity matrices. This gives rise to a necessary condition for the join of two (possibly disconnected) graphs $G$ and $H$ to be the pattern of an orthogonal symmetric matrix, or equivalently, for the minimum number of distinct eigenvalues $q$ of $G\vee H$ to be equal to two. Under additional hypotheses, we show that this necessary condition is also sufficient. As an application, we prove that $q(G\vee H)$ is either two or three when $G$ and $H$ are unions of complete graphs, and we characterise when each case occurs.