Signed graphs whose spectrum is bounded by −2

Abstract

We prove that for every tree T with t vertices (t>2), the signed line graph L(Kt) has L(T) as a star complement for the eigenvalue −2; in other words, T is a foundation for Kt (regarded as a signed graph with all edges positive). In fact, L(Kt) is, to within switching equivalence, the unique maximal signed line graph having such a star complement. It follows that if t∉{7,8,9} then, to within switching equivalence, Kt is the unique maximal signed graph with T as a foundation. We obtain analogous results for a signed unicyclic graph as a foundation, and then provide a classification of signed graphs with spectrum in [−2,∞). We note various consequences, and review cospectrality and strong regularity in signed graphs with least eigenvalue ≥−2.