Edge ideals with almost maximal finite index and their powers

A graded ideal $I$ in $\mathbb{K}[x_1,\ldots,x_n]$, where $\mathbb{K}$ is a field, is said to have almost maximal finite index if its minimal free resolution is linear up to the homological degree $\mathrm{pd}(I)-2$, while it is not linear at the homological degree $\mathrm{pd}(I)-1$, where $\mathrm{pd}(I)$ denotes the projective dimension of $I$. In this paper we classify the graphs whose edge ideals have this property. This in particular shows that for edge ideals the property of having almost maximal finite index does not depend on the characteristic of $\mathbb{K}$. We also compute the non-linear Betti numbers of these ideals. Finally, we show that for the edge ideal $I$ of a graph $G$ with almost maximal finite index, the ideal $I^s$ has a linear resolution for $s\geq 2$ if and only if the complementary graph $\bar{G}$ does not contain induced cycles of length $4$.