Jump Sequences of Edge Ideals

Given an edge ideal of graph G, we show that if the first nonlinear strand in the resolution of $I_G$ is zero until homological stage $a_1$, then the next nonlinear strand in the resolution is zero until homological stage $2a_1$. Additionally, we define a sequence, called a \emph{jump sequence}, characterizing the highest degrees of the free resolution of the edge ideal of G via the lower edge of the Betti diagrams of $I_G$. These sequences strongly characterize topological properties of the underlying Stanley-Reisner complexes of edge ideals, and provide general conditions on construction of clique complexes on a fix set of vertices. We also provide an algorithm for obtaining a large class of realizable jump sequences and classes of Gorenstein edge ideals achieving high regularity.