Castelnuovo-Mumford regularity of the closed neighborhood ideal of a graph

Abstract

Let \(G\) be a finite simple graph, and \(NI(G)\) denote the closed neighborhood ideal of \(G\) in a polynomial ring \(R\). We show that if \(G\) is a forest, then the Castelnuovo-Mumford regularity of \(R/NI(G)\) is the same as the matching number of \(G\), thus proving a conjecture of Sharifan and Moradi in the affirmative. We also show that the matching number of \(G\) provides a lower bound for the Castelnuovo-Mumford regularity of \(R/NI(G)\) when \(G\) is a chordal graph, complement of a tree, complete bipartite graph, cycle graph, or a wheel graph. Moreover, we investigate the relationship between these two invariants for two graph operations, namely, the join and the corona product of graphs.