Signed graphs whose spectrum is bounded by −2

Citation

Rowlinson P & Stanić Z (2022) Signed graphs whose spectrum is bounded by −2. Applied Mathematics and Computation, 423, Art. No.: 126991. https://doi.org/10.1016/j.amc.2022.126991

Abstract
We prove that for every tree T with t vertices (t > 2), the signed line graph ℒ (Kt) has ℒ (T) as a star complement for the eigenvalue -2; in other words, T is a foundation for Kt (regarded as a signed graph with all edges positive). In fact, ℒ (Kt) is, to within switching equivalence, the unique maximal signed line graph having such a star complement. It follows that if t ∉ {7, 8, 9} then, to within switching equivalence, Kt is the unique maximal signed graph with T as a foundation. We obtain analogous results for a signed unicyclic graph as a foundation, and then provide a classification of signed graphs with spectrum in [-2, ∞). We note various consequences, and review cospectrality and strong regularity in signed graphs with least eigenvalue ≥, -2.

Keywords
Adjacency matrix; Foundation of a signed graph; Signed line graph; Star complement; Star partition

Journal
Applied Mathematics and Computation: Volume 423