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arXiv · 1802.07202

Regularity of bicyclic Graphs and their powers

Yairon Cid-Ruiz·Sepehr Jafari·Navid Nemati·Beatrice Picone
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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.

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Yairon Cid-Ruiz, Sepehr Jafari, Navid Nemati, Beatrice Picone. 2018-02-20. Regularity of bicyclic Graphs and their powers. https://arxiv.org/abs/1802.07202

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