We study the asymptotic behaviour of v-number and local v-numbers of Noetherian generalized symbolic power filtrations \({\mathcal {I}}=\{I_n\}\) in a Noetherian \({\mathbb N}\) -graded domain and show that they are quasi-linear type. We provide sufficient conditions for the existence of the limits \(\lim \limits _{n\rightarrow \infty }\frac{v(I_n)}{n}\) and \(\lim \limits _{n\rightarrow \infty }\frac{v_{\mathfrak {p}}(I_n)}{n}\) for all \({\mathfrak {p}}\in {\overline{A}}({\mathcal {I}})\) . We explicitly compute local v-numbers and v-numbers of symbolic powers of cover ideals of complete bipartite graphs, complete graphs, cycles, \(K_m^s\) and compare them with their Castelnuovo–Mumford regularity. For every positive integer \(p\ge 2\) , we provide an example of an unmixed bipartite graph \({\mathcal {H}}_p\) that is not a complete multipartite graph and \(v(J({\mathcal {H}}_p))\ge \operatorname {bight}(I({\mathcal {H}}_p))\) . This answers a question of Saha in (Int Math Res Not IMRN 11:9010–9019, 2024, Question 3.12). We show that for both connected bipartite graphs and connected non-bipartite graphs, the difference between the regularity and the v-number of the cover ideals can be arbitrarily large. This strengthens and gives an alternative proof of Saha in (Int Math Res Not IMRN 11:9010-9019, 2024, Theorem 3.10).
