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Signed graphs whose spectrum is bounded by −2

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  • Rowlinson, Peter
  • Stanić, Zoran

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.

Suggested Citation

  • Rowlinson, Peter & Stanić, Zoran, 2022. "Signed graphs whose spectrum is bounded by −2," Applied Mathematics and Computation, Elsevier, vol. 423(C).
    • Handle: RePEc:eee:apmaco:v:423:y:2022:i:c:s0096300322000777
    DOI: 10.1016/j.amc.2022.126991
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    References listed on IDEAS

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    1. Hou, Yaoping & Tang, Zikai & Wang, Dijian, 2019. "On signed graphs with just two distinct Laplacian eigenvalues," Applied Mathematics and Computation, Elsevier, vol. 351(C), pages 1-7.
    2. Yuan, Xiying & Mao, Yanqi & Liu, Lele, 2021. "Maximal signed graphs with odd signed cycles as star complements," Applied Mathematics and Computation, Elsevier, vol. 408(C).
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