Signed graphs with three eigenvalues: Biregularity and beyond

Abstract

First we investigate net-biregular signed graphs with spectrum of the form [ρ,μm,λl] where λ is non-main; such graphs are necessarily biregular with exactly two main eigenvalues. We provide two constructions of signed graphs with three eigenvalues, where the graphs that arise include net-biregular and net-regular signed graphs having spectrum [ρ,μ,λl], with λ non-main. Secondly we determine all the connected signed graphs with spectrum [ρ,μ2,λl](l≥2) where λ is non-main: these include a new infinite family of signed graphs which are neither net-regular nor net-biregular. Thus, in contrast to the situation for graphs, a signed graph with two main eigenvalues and one non-main eigenvalue is not necessarily net-biregular.