Binomial Edge Ideals of Unicyclic Graphs

Let $G$ be a connected simple graph on the vertex set $[n]$. Banerjee-Betancourt proved that $depth(S/J_G)\leq n+1$. In this article, we prove that if $G$ is a unicyclic graph, then the depth of $S/J_G$ is bounded below by $n$. Also, we characterize $G$ with $depth(S/J_G)=n$ and $depth(S/J_G)=n+1$. We then compute one of the distinguished extremal Betti numbers of $S/J_G$. If $G$ is obtained by attaching whiskers at some vertices of the cycle of length $k$, then we show that $k-1\leq reg(S/J_G)\leq k+1$. Furthermore, we characterize $G$ with $reg(S/J_G)=k-1$, $reg(S/J_G)=k$ and $reg(S/J_G)=k+1$. In each of these cases, we classify the uniqueness of extremal Betti number of these graphs.