Castelnuovo-Mumford regularity of the closed neighborhood ideal of a graph
arxiv(2024)
摘要
Let G be a finite simple graph and let 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, unicyclic graph, complete bipartite graph, or the wheel graph. For
forests and unicyclic graphs, we show that the projective dimension of
R/NI(G) is also bounded below by the matching number of G. Moreover, we
investigate the relationship between the regularity and the matching number for
two graph operations, namely, the join and the corona product of two graphs.
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