On the equations of the edge cone of a graph and some applications
Let \(G\) be a graph on the vertex set \(V=\{\nu_1,\dots,\nu_n\}\) and \(R=k[x_1,\dots,x_n]\) the polynomial ring over a field \(k\). The monomial subring of \(G\) is the \(k\)-subalgebra \(k[G]=k[\{x_ix_j\mid\nu_i\) is adjacent to \(\nu_j\}]\) and the edge cone of \(G\) is the cone in \(\mathbb{R}^n\) spanned by the set of vectors \(e_i+e_j\) such that \(\nu_i\) is adjacent to \(\nu_j\) where \(e_i\) is the \(i\)-th unit vector. Suppose that \(G\) is a graph none of whose components is bipartite. This paper presents a combinatorial description of the faces of maximal dimension of the edge cone when \(k[G]\) has dimension \(n\). Estimates are given for the \(a\)-invariant of \(k[G]\) when \(G\) is the join of two normal (i.e., \(k[G]\) is normal) connected non bipartite graphs or \(G\) is the cone of a normal connected non bipartite graph. This paper generalizes work from \textit{W. Bruns}, \textit{W. V. Vasconcelos} and \textit{R. H. Villarreal} [Ill. J. Math. 41, No. 3, 341-353 (1997; Zbl 0908.13014)].
scientific article; zbMATH DE number 1240867