The $\mathrm{v}$-Number of Binomial Edge Ideals

The invariant $\mathrm{v}$-number was introduced very recently in the study of Reed-Muller-type codes. Jaramillo and Villarreal (J Combin. Theory Ser. A 177:105310, 2021) initiated the study of the $\mathrm{v}$-number of edge ideals. Inspired by their work, we take the initiation to study the $\mathrm{v}$-number of binomial edge ideals in this paper. We discuss some properties and bounds of the $\mathrm{v}$-number of binomial edge ideals. We explicitly find the $\mathrm{v}$-number of binomial edge ideals locally at the associated prime corresponding to the cutset $\emptyset$. We show that the $\mathrm{v}$-number of Knutson binomial edge ideals is less than or equal to the $\mathrm{v}$-number of their initial ideals. Also, we classify all binomial edge ideals whose $\mathrm{v}$-number is $1$. Moreover, we try to relate the $\mathrm{v}$-number with the Castelnuvo-Mumford regularity of binomial edge ideals and give a conjecture in this direction.