Regularity, matchings and Cameron-Walker graphs

Let $G$ be a simple graph and let $\nu (G)$ be the matching number of $G$. It is well-known that $\text{reg}I(G)⩽\nu (G)+1$. In this paper we show that $\text{reg}I(G)=\nu (G)+1$ if and only if every connected component of $G$ is either a pentagon or a Cameron-Walker graph.