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arXiv · 1710.01710

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
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Abstract

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 some classes of graphs.

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L. Emilio Allem, Antonio Cafure, Ezequiel Dratman, Luciano N. Grippo, Martín D. Safe, Vilmar Trevisan. 2017-10-04. Partial characterization of graphs having a single large Laplacian eigenvalue. https://arxiv.org/abs/1710.01710

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