Resolution and Betti numbers of Vertex cover ideals

The vertex cover ideal $J(G)$ of a finite graph $G$ is studied. We characterize when a Cohen–Macaulay vertex cover ideal $J(G)$ has a Scarf minimal free resolution. Furthermore, by using both combinatorial and topological techniques, the graded Betti number ${\beta }_{i,i+j}(J(G))$, 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.