  In this paper, we give formulas for $v$-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 $v$-number
of $I$, denoted by $v(I)$, and give an upper bound of $v(I)$ in terms of the
degree of its generators. We show that for a monomial ideal $I$, $v(I^{n+1})$
is bounded above by a linear polynomial for large $n$ and for certain classes
of monomial ideals, the upper bound is achieved for all $n\geq 1$. For
$\mathfrak m$-primary monomial ideal $I$ we prove that $v(I)\leq$ reg$(S/I)$
and their difference can be arbitrarily large.
