On the $\mathrm{v}$-number of binomial edge ideals of some classes of graphs

Let $G$ be a finite simple graph, and $J_G$ denote the binomial edge ideal of $G$. In this article, we first compute the $\mathrm{v}$-number of binomial edge ideals corresponding to Cohen-Macaulay closed graphs. As a consequence, we obtain the $\mathrm{v}$-number for paths. For cycle and binary tree graphs, we obtain a sharp upper bound for $\mathrm{v}(J_G)$ using the number of vertices of the graph. We characterize all connected graphs $G$ with $\mathrm{v}(J_G)=2$. We show that for a given pair $(k,m),k\le m$, there exists a graph $G$ with an associated monomial edge ideal $I$ having $\mathrm{v}$-number equal to $k$ and regularity $m$. If $2k\le m$, then there exists a binomial edge ideal with $\mathrm{v}$-number $k$ and regularity $m$. Finally, we compute $\mathrm{v}$-number of powers of binomial edge ideals with linear resolution, thus proving a conjecture on the $\mathrm{v}$-number of powers of a graded ideal having linear powers, for the class of binomial edge ideals.