A study of \({\textrm{v}}\)-number for some monomial ideals

In this paper, we give formulas for \({\textrm{v}}\) $\text{v}$ -number of edge ideals of some graphs like path, cycle, 1-clique sum of a path and a cycle, 1-clique sum of two cycles and join of two graphs. For an \({\mathfrak {m}}\) $\mathfrak{m}$ -primary monomial ideal \(I\subset S=K[x_1,\ldots ,x_t]\) $I\subset S=K[x_1,\dots ,x_t]$ , we provide an explicit expression of \({\textrm{v}}\) $\text{v}$ -number of I, denoted by \({\textrm{v}}(I)\) $\text{v}(I)$ , and give an upper bound of \({\textrm{v}}(I)\) $\text{v}(I)$ in terms of the degree of its generators. We show that for a monomial ideal I, \({\textrm{v}}(I^{n+1})\) $\text{v}(I^{n+1})$ is bounded above by a linear polynomial for large n and for certain classes of monomial ideals, the upper bound is achieved for all \(n\ge 1\) $n\ge 1$ . For \({\mathfrak {m}}\) $\mathfrak{m}$ -primary monomial ideal I we prove that \({\textrm{v}}(I)\le {\text {reg}}(S/I)\) $\text{v}(I)\le \text{reg}(S/I)$ and their difference can be arbitrarily large.