The slope of v-function and Waldschmidt Constant

In this paper, we study asymptotic behaviour of the v-numbers of a Noetherian filtration I= \I₊\₊ ₀ of ideals in a Noetherian graded domain R. Recently, it is shown that v (I₊) is periodically linear in k for k>>0. We show that ₊ v (I₊) k exists and ₊ v (I₊) k= ₊ (I₊) k. That is, all these linear functions have same slope, which is equal to ₊ (I₊) k. In particular, for Noetherian symbolic filtration, we have ₊ v (I^ (k) ) k= (I), the Waldschmidt constant of I. Also, for several classes of square-free monomial ideals, we show that v (I^ (k) ) reg (R/I^ (k) ) for all k 1. As a special case, for any simple graph G, we show that v (J (G) ^ (k) ) reg (R/J (G) ^ (k) ) for all k 1 and v (J (G) ^ (k) ) = reg (R/J (G) ^ (k) ) = (J (G) ^ (k) ) -1 for all k 1 if and only if G is a Cohen-Macaulay very-well covered graph, where J (G) denotes the cover ideal of G.