On the v-number of binomial edge ideals of some classes of graphs

Let G be a finite simple graph, and [Formula: see text] denote the binomial edge ideal of G. In this paper, we first compute the [Formula: see text]-number of binomial edge ideals corresponding to Cohen–Macaulay closed graphs. As a consequence, we obtain the [Formula: see text]-number for paths. For cycle and binary tree graphs, we obtain a sharp upper bound for [Formula: see text] using the number of vertices of the graph. We characterize all connected graphs G with [Formula: see text]. We show that for a given pair [Formula: see text], there exists a graph G with an associated monomial edge ideal I having [Formula: see text]-number equal to k and regularity m. We also show that if [Formula: see text], then there exists a binomial edge ideal with [Formula: see text]-number k and regularity m. Finally, we compute [Formula: see text]-number of powers of binomial edge ideals with linear resolution, thus proving a conjecture on the [Formula: see text]-number of powers of a graded ideal having linear powers, for the class of binomial edge ideals.