  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 $\pd(R/I(G))=\max_{v\in V(G)}
\{\deg_G(v)\}$.
