Binomial edge ideals of unicyclic graphs

Let G be a connected graph on the vertex set [n]. Then depth(S/J(G)) <= n + 1. In this paper, we prove that if G is a unicyclic graph, then the depth of S/JG 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 <= reg(S/JG) <= 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 the extremal Betti number of these graphs.