Signed graphs with exactly two distinct main eigenvalues

Abstract

An eigenvalue λ of a signed graph S of order n is a main eigenvalue if its eigenspace is not orthogonal to the all-ones vector j. Characterizing signed graphs with exactly k(1≤k≤n) main eigenvalues is a problem in algebraic and graph theory that has been studied since 2020. Z. Stanić has noticed that a signed graph has exactly one main eigenvalue if and only if it is net-regular, and in this paper, we study signed graphs with exactly two distinct main eigenvalues by studying (0,1,2)-multi-graphs with exactly two distinct main eigenvalues. We partially characterize the case when the basic graphs of the (0,1,2)-multi-graphs are trees, and propose some problems for further research.