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2 results

  • 2 - Eigenvalues

  • Zoran Stanić, University of Belgrade, Serbia
  • Book:
    Spectra of Signed Graphs
    Published online:
    15 May 2026
    Print publication:
    11 June 2026, pp 22-63

  • Summary

    This chapter focuses on the eigenvalues of a prescribed matrix associated with a signed graph. The exposition begins with the concept of main eigenvalues and examines their role in various structural analyses. It then proceeds to specific computations, including determinants, relationships with the spectrum of the underlying graph, and the effects of particular operations on signed graphs and their resulting spectra. Further developments consider signed graphs with specified spectral properties, such as those exhibiting a symmetric spectrum. The chapter also explores eigenvalue bounds, including limit points for the largest eigenvalue, spectral distances between signed graphs, and spectral deviations of a signed graph relative to related signed graphs, providing a detailed and comprehensive analysis of these spectral characteristics.

  • Hoffman program of Laplacian matching polynomials of graphs

  • Part of
  • Zhaoxi Li, Jianfeng Wang, Shi-Mei Ma, Francesco Belardo
  • Journal:
    Proceedings of the Edinburgh Mathematical Society / Volume 69 / Issue 2 / May 2026
    Published online by Cambridge University Press:
    10 November 2025, pp. 581-598
    • Article
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  • Let $\mathcal{G}$ be the class of all connected simple graphs. The Hoffman program of graphs with respect to a spectral invariant $\lambda(G)$ consists of determining all the limit points of the set $\{\lambda(G)\,\vert\, G\in\mathcal{G}\}$ and characterising all $G$’s such that $\lambda(G)$ does not exceed a fixed limit point. In this paper, we study the Hoffman program for Laplacian matching polynomials of graphs in regard to their largest Laplacian matching roots. Precisely, we determine all the limit points of the largest Laplacian matching roots of graphs less than $\tau = 2+\omega^{\frac{1}{2}}+\omega^{-\frac{1}{2}}(=4.38+)$, and then characterise the connected graphs with the largest Laplacian matching roots less than $2+\sqrt{5}$, where $\omega=\frac{1}{3}(\sqrt[3]{19+3\sqrt{33}}+\sqrt[3]{19-3\sqrt{33}}+1)$.