Star complements in signed graphs with two symmetric eigenvalues

Abstract

We consider signed graphs $G$ whose spectrum is comprised of exactly two (distinct) eigenvalues that differ only in sign, abbreviated to signed graphs with two symmetric eigenvalues. We obtain some relationships between such signed graphs and their star complements. Our results include structural examinations and constructions of infinite families of signed graphs with two symmetric eigenvalues. We also determine the bases for the eigenspaces of the eigenvalues of $G$ in terms of the eigenspaces of its star complement. In particular, we consider the case in which a star complement has two symmetric eigenvalues, as well.