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

Abstract

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