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

Abstract

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 $a^{\prime }(G)$ 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 $\pd(R/I(G))=\max_{v\in V(G)} \{°_G(v)\}$.

arXiv comment11 pages

References in corpus (2)

• Pretty clean monomial ideals and linear quotients
• Algebraic properties of edge ideals via combinatorial topology

Cited by in corpus (1)

• Shedding vertices and Ass-decomposable monomial ideals