Resolution and Betti numbers of Vertex cover ideals

The vertex cover ideal $J(G)$ of a finite graph G is studied. It is known that $J(G)$ is Cohen--Macauly if and only if the complementary graph of G is a chordal graph. In this paper, we show that $J(G)$ is Gorenstein if and only if G is a complete bipartite graph. Also, when a Cohen--Macaulay vertex cover ideal $J(G)$ has a Scarf resolution is characterized. Furthermore, by using both combinatorial and topological techniques, the graded Betti number βi,i+j(J(G)), where i and j are the projective dimension and the regularity of $J(G)$, is computed, when G is either a path or a cycle.