On the equations of the edge cone of a graph and some applications
type: Item
instance of: scholarly article
title: On the equations of the edge cone of a graph and some applications
zbMATH Open document ID: 0920.13015
author: Rafael H. Villarreal
published in: Manuscripta Mathematica
publication date: 1999-09-13T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 52B05
Mathematics Subject Classification ID: 05C50
zbMATH DE Number: 1240867
zbMATH Keywords: face of edge cone
zbMATH Keywords: \(a\)-invariant
zbMATH Keywords: monomial subring
reviewed by: D. D. Anderson
MaRDI profile type: Publication
full work available at URL: s002290050103
OpenAlex ID: W1969459276
DOI: 10.1007/S002290050103
description: scientific article; zbMATH DE number 1240867