Connected non-complete signed graphs which have symmetric spectrum but are not sign-symmetric

Abstract

A signed graph Ġ is called sign-symmetric if it is switching isomorphic to its negation −Ġ, where −Ġ is obtained by reversing the sign of every edge of Ġ. The authors of Belardo et al. (2018) constructed a complete signed graph that is not sign-symmetric, but has a symmetric spectrum and posted the following problem: Are there connected non-complete signed graphs whose spectrum is symmetric but they are not sign-symmetric? In this paper we positively address this problem. Our examples include infinite families constructed on the basis of the Cartesian product and the corona product of signed graphs. We note that the same problem was first resolved in Ghorbani et al. (2020) by means of different constructions.