  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 $\mathrm{im}(G)$ and the regularity $\mathrm{reg}(R/I(G))$ of a
graph $G$, satisfy $\mathrm{v}(I(G))\leq \mathrm{im}(G)\leq
\mathrm{reg}(R/I(G))$, where $G$ is either a bipartite graph, or a
$(C_{4},C_{5})$-free vertex decomposable graph, or a whisker graph. There is an
open problem in \cite{v}, whether $\mathrm{v}(I)\leq \mathrm{reg}(R/I)+1$ for
any square-free monomial ideal $I$. We show that
$\mathrm{v}(I(G))>\mathrm{reg}(R/I(G))+1$, 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 $\mathrm{reg}(R/I(G))$ 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.
