Signed graphs with three eigenvalues: Biregularity and beyond

Citation

Rowlinson P & Stanić Z (2021) Signed graphs with three eigenvalues: Biregularity and beyond. Linear Algebra and its Applications, 621, pp. 272-295. https://doi.org/10.1016/j.laa.2021.03.018

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.

Keywords
Adjacency matrix; Biregular graph; Block design; Graph spectrum; Net-biregular signed graph; Star complement

Journal
Linear Algebra and its Applications: Volume 621