of index $>1$ can be characterized by the fact that certain graphs associated with such ideals are connected. This criterion is used in the proof of Theorem \[main\].

If the index of a graded ideal is finite, then it is at most its projective dimension. In the case that $\index(I)=\projdim(I)$ we say that $I$ has maximal finite index. In Section  3 edge ideals of maximal finite index are classified. They turn out to be the edge ideals of the complement of a cycle, see Theorem \[main2\]. The essential tools to prove this result are Hochster’s formula to compute the graded Betti numbers of a squarefree monomial ideal as well as the result of [@EGHP Theorem 2.1] in which the index of an edge ideal is characterized in terms of the underlying graph. As a consequence of Theorem \[main2\] it is shown in Corollary \[maxlinear\] that all powers $I^k$ for $k\geq 2$ have a linear resolution for an ideal of maximal finite index $>1$. This supports our conjecture that the index of the powers $I^k$ of an edge ideal $I$ is a strictly increasing function on $k$.

Our final Section 4 is devoted to the study of the index of the 