for any $k>0$. Moreover, by using Theorem \[main\], if $\index(I)=1$, then $\index(I^k)=1$ for any $k>0$.

\(b) Let $G$ be a simple graph on the vertex set $\{x_{1}, x_{2}, \ldots, x_{n}\}$. We define the [*whisker graph*]{} $W(G)$ of $G$ by adding to the vertex set of $G $ the new vertices $\{y_1, y_2, \ldots, y_n\}$. The edge set of $W(G)$ is $E(G)\cup \{\{x_{i}, y_{i}\}: 1\leq i\leq n\}$. Furthermore by $ L(G)$ we denote a graph obtained from $G$ by adding a loop to each of its vertices and call it the [*loop graph*]{} of $G$.

Again as a consequence of Theorem \[main\], it follows together with [@HHZ2 Theorem 3.2] that $I(L(G))^k$ has a linear resolution for all $k$ if and only if $G$ is complete, and $\index (I(L(G))^k)=1$ for all $k$ if and only if $G$ is not complete. A similar statement holds for $I(W(G))$, because $I(W(G))$ is obtained from $I(L(G))$ by polarization.

Edge ideals of maximal finite index
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A monomial ideal $I$ of finite index has $\index(I)\leq \projdim(I)$. We say that $I$ has [*maximal finite index*]{} if equality holds. In this section we classify those graphs whose edge ideal has maximal finite index. In particular our aim is to 