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 ₈, ₈+₉ (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.