Partial characterization of graphs having a single large Laplacian eigenvalue

L. EMILIO ALLEM; ANTONIO A. CAFURE; EZEQUIEL DRATMAN; LUCIANO N. GRIPPO; MARTÍN D. SAFE; VILMAR TREVISAN

ELECTRONIC JOURNAL OF COMBINATORICS, THE

ELECTRONIC JOURNAL OF COMBINATORICS

Año: 2018 vol. 25

1077-8926

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.