Regularity of powers of edge ideals
Description
Let $G$ be a finite simple graph and let $I = I(G)$ be its edge ideal. Main goal in this thesis is to relate algebraic invariants of powers of edge ideals and combinatorial data of graphs. In particular, we focus on the Castelnuovo-Mumford regularity of an edge ideal and its powers. The first part of this thesis focuses on regularity of edge ideals. In that regard, we give new bounds on the regularity of $I$ when $G$ contains a Hamiltonian path and when $G$ is a Hamiltonian graph. Moreover, we explicitly compute the regularity of unicyclic graphs and characterize the unicyclic graphs with regularity $\nu(G)+1$ and $\nu(G)+2$ where $\nu(G)$ denotes the induced matching number of $G.$ The second problem is on the regularity of powers of edge ideals. Let $R=k[x_1, \ldots, x_n]$ be a polynomial ring and let $I \subset R$ be a homogeneous ideal. It is a celebrated result of Cutkosky, Herzog,Trung \cite{CHT}, Kodiyalam \cite{Kodi}, Trung and Wang \cite{TW} that regularity of $I^s$ is asymptotically a linear function for $s \gg 0,$ i.e., $as+b$ for integers $a,b$ and $s_0$ when $s \geq s_0.$ It is known that $a$ is equal to 2 when $I=I(G)$ is the edge ideal of a graph. We then turn on our focus on identifying $b$ and $s_0$ via combinatorial data of the graph $G.$ We explicitly compute the regularity of $I^s$ for all $s\geq 1$ when $G$ is a forest, a cycle and a unicyclic graph. We also present a lower bound on the regularity of powers of edge ideals in terms of the induced matching number of a graph.