  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.
