The $v$-number of generalized binomial edge ideals of some graphs

Let $G$ be a finite connected simple graph, and let $\mathcal{J}_{K_m,G}$ denote its generalized binomial edge ideal. By investigating the colon ideals of $\mathcal{J}_{K_m,G}$, we derive a formula for the local $\mathrm{v}$-number of $\mathcal{J}_{K_m,G}$ with respect to the empty cut set. Furthermore, we classify graphs for which this generalized binomial edge ideal has $\mathrm{v}$-numbers $1$ or $2$. When $G$ is a connected closed graph, we compute the local $\mathrm{v}$-number of $\mathcal{J}_{K_2,G}$ by generalizing the work of Dey et al. Additionally, under the condition that $G$ is Cohen--Macaulay, we derive formulas for the $\mathrm{v}$-number of $\mathcal{J}_{K_m,G}$ and $\mathcal{J}_{K_2,G}^k$, and show that the $\mathrm{v}$-number of $\mathcal{J}_{K_2,G}^k$ is a linear function of $k$.