We prove that for every tree T with t vertices ( t > 2 ), the signed line graph (K-t) pound has (T) pound as a star complement for the eigenvalue -2 ; in other words, T is a foundation for K-t (regarded as a signed graph with all edges positive). In fact, (K-t) pound is, to within switching equivalence, the unique maximal signed line graph having such a star complement. It follows that if t is not an element of{7, 8, 9 } then, to within switching equivalence, K-t 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, infinity). We note various consequences, and review cospectrality and strong regularity in signed graphs with least eigenvalue >=-2. (c) 2022 Elsevier Inc. All rights reserved.
Adjacency matrixFoundation of a signed graphSigned line graphStar complementStar partition
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