Regularity of $3$-Path Ideals of Trees and Unicyclic Graphs

Let $G$ be a simple graph and $I_3(G)$ be its $3$-path ideal in the corresponding polynomial ring $R$. In this article, we prove that for an arbitrary graph $G$, $reg(R\mathrm{/}{I}_3(G))$ is bounded below by $2{\nu }_3(G)$, where ${\nu }_3(G)$ denotes the $3$-path induced matching number of $G$. We give a class of graphs, namely, trees for which the lower bound is attained. Also, for a unicyclic graph $G$, we show that $reg(R\mathrm{/}{I}_3(G))\le 2{\nu }_3(G)+2$ and provide an example that shows that the given upper bound is sharp.