The $\mathrm{v}$-number of Monomial Ideals

We generalize some results of $\mathrm{v}$-number for arbitrary monomial ideals by showing that the $\mathrm{v}$-number of an arbitrary monomial ideal is the same as the $\mathrm{v}$-number of its polarization. We prove that the $\mathrm{v}$-number $\mathrm{v}(I(G))$ of the edge ideal $I(G)$, the induced matching number $\mathrm{i}\mathrm{m}(G)$ and the regularity $\mathrm{r}\mathrm{e}\mathrm{g}(R\mathrm{/}I(G))$ of a graph $G$, satisfy $$\mathrm{v}$(I(G))\leq $\mathrm{i}\mathrm{m}$(G)\leq $\mathrm{r}\mathrm{e}\mathrm{g}$(R/I(G))$,where$G$iseitherabipartitegraph,ora$(C_{4},C_{5})$-free vertex decomposable graph, or a whisker graph. There is an open problem in \cite{v}, whether $\mathrm{v}(I)\le \mathrm{r}\mathrm{e}\mathrm{g}(R\mathrm{/}I)+1$ for any square-free monomial ideal $I$. We show that $\mathrm{v}(I(G))>\mathrm{r}\mathrm{e}\mathrm{g}(R\mathrm{/}I(G))+1$, for a disconnected graph $G$. We derive some inequalities of $\mathrm{v}$-numbers which may be helpful to answer the above problem for the case of connected graphs. We connect $\mathrm{v}(I(G))$ with an invariant of the line graph $L(G)$ of $G$. For a simple connected graph $G$, we show that $\mathrm{r}\mathrm{e}\mathrm{g}(R\mathrm{/}I(G))$ can be arbitrarily larger than $\mathrm{v}(I(G))$. Also, we try to see how the $\mathrm{v}$-number is related to the Cohen-Macaulay property of square-free monomial ideals.