Signed graphs whose spectrum is bounded by −2



Rowlinson P & Stanić Z (2022) Signed graphs whose spectrum is bounded by −2. Applied Mathematics and Computation, 423, Art. No.: 126991.

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.

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

Applied Mathematics and Computation: Volume 423

Publication date15/06/2022
Publication date online14/02/2022
Date accepted by journal26/01/2022

People (1)


Professor Peter Rowlinson

Professor Peter Rowlinson

Emeritus Professor, Mathematics