Gwyneth Whieldon

This thesis compiles results in four related areas.

• Jump Sequences of Edge Ideals: Given a graph G on n vertices with edge ideal I(G), we introduce a new invariant Jump(I(G)) which describes the possible Betti tables of I(G). We show that the smallest k such that β_k,k+3(I(G)) =/= 0 is bounded below in terms of smallest j such that β_j,j+2(I(G)) =/= 0. In addition, we show that for ideals I(G) such that β_2,4(I(G)) = 0 and fewer than 11 vertices satisfy reg(I(G)) ≤ 3. We construct large classes of examples partially spanning the set of Betti tables of I(G) with reg(I(G)) = k.
• Stabilization of Betti Tables: Let R be a polynomial ring. Given a homogeneous ideal I ⊆ R equigenerated in degree r, we show that the Betti tables of I^d stabilize into a fixed shape for all d ≥ D for some D.
• Linear Quotients Ordering of Anticycle: Let A_n be the anticycle graph on n vertices and P_n be the antipath graph on n vertices. We produce a linear quotients ordering on all powers of the edge ideal of the antipath I(P_n)^k, and a linear quotients order on the second power of the edge ideal of P_n the anticycle I(A_n)².
• Nerve Complexes of Graphs: We examine the nerve complex N(G) of a graph G. We show that the Betti numbers of this complex encode spanning trees, matchings, genus, k-edge connectivity, and other invariants of G.