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

Abstract

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