Regularity of bicyclic Graphs and their powers

TL;DR

This paper characterizes the regularity of edge ideals of bicyclic graphs using induced matching numbers and provides explicit formulas for dumbbell graphs, revealing how regularity evolves with graph powers.

Contribution

It offers a characterization of the Castelnuovo-Mumford regularity for bicyclic graphs and explicitly computes the induced matching number for dumbbell graphs, including regularity of their powers.

Findings

Regularity of $I(G)$ relates to the induced matching number of $G$.

Explicit computation of induced matching number for dumbbell graphs.

Regularity of powers of $I(G)$ follows a linear formula for certain dumbbell graphs.

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 $ \operatorname{reg}(I(G)^q)=2q+\operatorname{reg}(I(G))-2$, for all $ q\geq 1 $, when $ G $ is a dumbbell graph with a connecting path having no more than two vertices.