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.

Paths are generically realisable

“…The maximum multiplicity of an eigenvalue of a path is one [11], so every multiplicity vector for P n is a 0-1 vector, hence it is generically realisable by Theorem 2.5. Complete graphs are also generically realisable [17], so the second assertion follows immediately.…”

“…If every multiplicity matrix for a graph G is generically realisable for G, then we say that G is generically realisable. This is a strong requirement for a graph; in particular, it implies that G is spectrally arbitrary for every multiplicity matrix that can be realised by G. In fact, the only families of generically realisable graphs previously known are unions of complete graphs [17]. We can now extend this to include paths.…”

“…The present work draws motivation from the above result, in the context of multiplicity matrices of disconnected graphs. Such multiplicity matrices were introduced in [17] as a generalisation of ordered multiplicity vectors of connected graphs, and are defined as follows. Let G be a graph with k connected components G 1 , .…”

“…Our main result is Theorem 2.5 below, in which we apply the SSP to show that every 0-1 multplicity matrix which fits a graph G is generically realisable for G. In particular, this implies that every multiplicity matrix which fits a path P n is generically realisable for P n . In the terminology of [17], we say P n is generically realisable.…”

“…where q(A) denotes the number of distinct eigenvalues of a square matrix A, is one of the parameters motivated by IEP-G. The study of q(G) was initiated by Leal-Duarte and Johnson in [15], and it has been broadly studied since then (see e.g., [2,3,[6][7][8]16,17,19]). Monfared and Shader [19,Theorem 5.2] proved that q(G ∨ H) = 2 if G and H are connected graphs with G = H .…”