Regularity of bicyclic Graphs and their powers

Abstract

Let $I(G)$ be the edge ideal of a bicyclic graph. In this paper, we characterize the Castelnuovo-Mumford regularity of $I(G)$ in terms of the induced matching number of $G$. For the base case of this family of graphs, i.e. dumbbell graph, we explicitly compute the induced matching number. Moreover, we prove that $\mathrm{reg}(I(G{)}^q)=2q+\mathrm{reg}(I(G))-2$, for all $q\ge 1$, when $G$ is a dumbbell graph with a connecting path having no more than two vertices.