  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.
