  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.
