Partial Characterization of Graphs Having a Single Large Laplacian Eigenvalue

  • L. Emilio Allem
  • Antonio Cafure
  • Ezequiel Dratman
  • Luciano N. Grippo
  • Martín D. Safe
  • Vilmar Trevisan
Keywords: Graphs, Laplacian matrix, Laplacian eigenvalues


The parameter $\sigma(G)$ of a graph $G$ stands for the number of Laplacian eigenvalues greater than or equal to the average degree of $G$. In this work, we address the problem of characterizing those graphs $G$ having $\sigma(G)=1$. Our conjecture is that these graphs are stars plus a (possible empty) set of isolated vertices. We establish a link between $\sigma(G)$ and the number of anticomponents of $G$. As a by-product, we present some results which support the conjecture, by restricting our analysis to cographs, forests, and split graphs.

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