Orthogonal symmetric matrices and joins of 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∨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∨H) is either two or three when G and H are unions of complete graphs, and we characterise when each case occurs.