V(G_{I^k}^{(u, v)})$ such that $\deg (\lcm(u,w))=2k+1$. Clearly, $w\in V(G_{I^k}^{(u, v)})$ yields that $w\in G(I^k)$ with the property that $w$ divides $\lcm(u,v)$. Moreover, $\deg (\lcm(u,w))=2k+1$ implies that either $x_i^kx_{i'}^{k-1}$ or $x_i^{k-1}x_{i'}^{k}$ divides $w$.

Note that $\lcm(u, v)=x_i^kx_{i'}^kx_{j}^kx_{j'}^k$ and since $w$ divides $\lcm(u,v)$, either $x_j$ or $x_{j'}$ divides $w$. This means that one of $\{i, j \}$, $\{i, j'\}$, $\{i', j\}$ or $\{i', j'\}$ must be an edge of $G$ which is a contradiction. Therefore, $u$ and $v$ are not connected in $G_{I^k}^{(u, v)}$. Corollary \[connected\] implies that $I^k$ does not have linear relations, a contradiction.

\(a) Let $G$ be a tree on the vertex set $[n]$ and $I$ its edge ideal. Then either $\index(I^k)=1$ or $\index(I^k)=\infty$ for any $k>0$. Indeed, suppose that $\index(I)=t<\infty$. If $n\leq 4$, Fröberg’s theorem [@F Theorem 1 ] implies that $\index(I)=\infty$. Therefore, $n>4$. By [@EGHP Theorem 2.1] there exists a minimal cycle $C$ of length $t+3$ in $\bar{G}$. Suppose that $V(C)=\{1, 2, \ldots, t+3\}$ and $E(C)=\{\{i, i+1\}\:\ 1\leq i\leq t+2\}\cup\{\{1, t+3\}\}$. If $t>1$, then $|V(C)|\geq 5$ and since $C$ is minimal we have $\{1,3\}, \{1,4\}, \{2,4\}, \{2,5\}, \{3,5\}\notin E(\bar{G})$. Therefore there exists a cycle in $G$, a contradiction.

By using [@HHZ2 Theorem 3.2] if $\index(I)=\infty$, then $\index(I^k)=\infty$ 