Projective Dimension of Some Graphs

In this paper exact values for the projective dimension of edge ideals associated to some star related graphs and product graphs $G\ \square\ P_2$, when $G=\ C_n,\ K_n$ and upper bounds for the projective dimension when $G=\ P_n,\ W_n$, are obtained. We have proved that $pd(C_{n+1}\ \square\ P_2)= 2\big(n-\left\lfloor \frac{n}{4}\right\rfloor\big)$, $pd(K_n\ \square\ P_2)= 2n-2$ and $pd(P_{n+1}\ \square \ P_2)\le n+3+\left\lfloor \frac{n-3}{2}\right\rfloor$, $pd(W_n\ \square\ P_2)\leq n+1+\lceil\frac{2n-1}{3}\rceil$. These values are functions of the number of vertices in the corresponding graphs.