Signed Toral Tessellations Whose Spectrum Consists of Exactly Two Symmetric Eigenvalues

Abstract

Signed graphs with exactly two eigenvalues constitute a rich and extensively studied class, yet they remain far from fully classified. Many structural properties are known and numerous families have been constructed, and any new non-trivial construction continues to offer notable progress. In this paper, the known family of signed graphs with eigenvalues 2 and −2, previously recognized as the 4-regular toral tessellations, is extended by the introduction of an infinite family of signed graphs whose spectra consist solely of two symmetric eigenvalues √λ and −√λ, where λ is an unbounded integer. Furthermore, a complete characterization of imposed signed toral tessellations is provided; one of its consequences is that every such graph necessarily has even order and even vertex degree.