Antonio Cafure

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.

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.

We determine conditions on $q$ for the nonexistence of deep holes of the standard Reed-Solomon code of dimension $k$ over $\mathbb F_q$ generated by polynomials of degree $k+d$. Our conditions rely on the existence of $q$-rational points with nonzero, pairwise-distinct coordinates of a certain family of hypersurfaces defined over $\mathbb F_q$. We show that the hypersurfaces under consideration are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning the singular locus of these hypersurfaces, from which the existence of $q$-rational points is established.