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In this paper, we explain the regularity, projective dimension and depth of edge ideal of some classes of graphs in terms of invariants of graphs. We show that for a C5-free vertex decomposable graph G, reg(R/I(G)) = cG, where cG is the maximum number of 3-disjoint edges in G. Moreover for this class of graphs we characterize pd(R/I(G)) and depth(R/I(G)). As a corollary we describe these invariants in forests and sequentially CohenMacaulay bipartite graphs.

Regularity and projective dimension of the edge ideal of $C_5$-free vertex decomposable graphs