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

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 $a^{\mathrm{\prime }}(G)$ and $n(G)$ depending on graph $G$ and show that for a vertex decomposable graph $G$, $\text{reg}(R\mathrm{/}I(G))\le \min \{a^{\mathrm{\prime }}(G),n(G)\}$ and for a shellable graph $G$, $\text{reg}(R\mathrm{/}I(G))\le n(G)$. Moreover it is shown that for a graph $G$, where $G^c$ is a $d$-tree, we have $\pd(R/I(G))=\max_{v\in V(G)} {\deg_G(v)}$.