Theoretic properties of the symmetric algebra of edge ideals

Abstract

In this note the integrality and the reducedness for the symmetric algebra of the edge ideals of simple graphs are analyzed. More precisely, we will extend a result on the integrality of the symmetric algebra given for connected graphs, stating a characterization for any simple graph. Moreover, through the Gröbner bases theory, we investigate simple graphs for which the symmetric algebra is reduced, in particular we prove its reducedness for graphs with $n$ vertices consisting of a cycle of length $n$.