Orthogonal symmetric matrices and joins of graphs

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