Signed graphs whose spectrum is bounded by −2

Peter Rowlinson and Zoran Stanić

Applied Mathematics and Computation, 2022, vol. 423, issue C

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 (search for similar items in EconPapers)
Date: 2022
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:423:y:2022:i:c:s0096300322000777

DOI: 10.1016/j.amc.2022.126991

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