Theoretic properties of the symmetric algebra of edge ideals (Q2349227)

Given a simple graph on \(n\) vertices, its edge ideal is defined as the ideal in a polynomial ring \(R=k[x_1,\dots,x_n]\) over a field, generated by all monomials \(x_ix_j\) corresponding to the edges \(\{v_i,v_j\}\) in the graph. The symmetric algebra of the graph is the quotient of a polynomial ring over \(R\), generated by a set of variables \(Y_{ij}\) indexed by the edges, modulo certain relations, given in terms of the graph structure. The paper characterizes the simple graphs for which the symmetric algebra is an integral domain as the graphs in which every connected subgraph has either at most one cycle of odd length or it has no cycle of even length. Using Gröbner basis theory, the paper also gives a sufficient condition for the symmetric algebra to be reduced. This sufficient condition is satisfied when the graph is a single cycle.