Non-vanishingness of Betti numbers of edge ideals
Kyouko Kimura
TL;DR
This paper investigates the conditions under which the graded Betti numbers of edge ideals of graphs do not vanish, providing characterizations for chordal graphs and forests, and analyzing projective dimension.
Contribution
It offers necessary and sufficient conditions for non-vanishing Betti numbers in edge ideals, specifically characterizing chordal graphs and forests, and determines the projective dimension for chordal graphs.
Findings
Characterization of non-vanishing Betti numbers for chordal graphs
Explicit description of Betti numbers for forests
Determination of projective dimension for chordal graphs
Abstract
Given finite simple graph one can associate the edge ideal. In this paper we discuss the non-vanishingness of the graded Betti numbers of edge ideals in terms of the original graph. In particular, we give a necessary and sufficient condition for a chordal graph on which the graded Betti number does not vanish and characterize the graded Betti number for a forest. Moreover we characterize the projective dimension for a chordal graph.
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Taxonomy
TopicsCommutative Algebra and Its Applications · Polynomial and algebraic computation · Algebraic structures and combinatorial models