Binomial edge ideals of Cameron-Walker graphs

Let $G$ be a Cameron--Walker graph on $n$ vertices and $J_G$ the binomial edge ideal of $G$. Let $S$ denote the polynomial ring in $2n$ variables over a field. It is shown that the following conditions are equivalent: (i) $S/J_G$ is Cohen--Macaulay; (ii) $J_G$ is unmixed; (iii) $\dim (S/J_G) = n+1$; (iv) (a) $n = 3$ and $G$ is a path of length $2$ or (b) $n = 5$ and $G$ is a path of length $4$ or (c) $n=5$ and $G$ is obtained by attaching a path of length $2$ to a triangle. Moreover, the depth of $S/J_G$ is computed for a class of Cameron--Walker graphs, called minimal dense Cameron--Walker graphs. As an application, it is proved that finite graphs $G$ with $\depth(S/J_G)=6$ can have any number of vertices~$n\geq 6$. Finally, it is shown that given integers $t,n$ with $6\leq t\leq n+1$, there exists a finite connected graph $G$ with $\depth (S/J_G)=t$.