Bounds for the regularity of edge ideal of vertex decomposable and shellable graphs

Details Bounds for the regularity of edge ideal of vertex decomposable and shellable graphs Bounds for the regularity of edge ideal of vertex decomposable and shellable graphs

2010-07-23

arxiv.org/abs/1007.4056v1

Mathematics

Commutative Algebra

11 pages

Scientific paper

In this paper we give upper bounds for the regularity of edge ideal of some classes of graphs in terms of invariants of graph. We introduce two numbers $a'(G)$ and $n(G)$ depending on graph $G$ and show that for a vertex decomposable graph $G$, $\reg(R/I(G))\leq \min\{a'(G),n(G)\}$ and for a shellable graph $G$, $\reg(R/I(G))\leq n(G)$. Moreover it is shown that for a graph $G$, where $G^c$ is a $d$-tree, we have $\pd(R/I(G))=\max_{v\in V(G)} \{\deg_G(v)\}$.

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