Some Results on $\mathrm{v}$-Number of Monomial Ideals

This paper investigates the v-number of various classes of monomial ideals. First, we considers the relationship between the v-number and the regularity of the mixed product ideal $I$, proving that $\mathrm{v}(I)\le \mathrm{r}\mathrm{e}\mathrm{g}(S\mathrm{/}I)$. Next, we investigate an open conjecture on the v-number: if a monomial ideal $I$ has linear powers, then for all $k\ge 1$, $\mathrm{v}(I^k)=\alpha (I)k-1.$ We prove that if a monomial ideal $I$ with linear powers is a homogeneous square-free ideal and ($k\ge 1$) has no embedded associated primes, then $\mathrm{v}(I^k)=\alpha (I)k-1.$ We have also drawn some conclusions about the k-th power of the graph.Additionally, we calculate the v-number of various powers of edge ideals(including ordinary power ,square-free powers, symbolic powers). Finally, we propose a conjecture that the v-number of ordinary powers of line graph is equal to the v-number of square-free powers.