We give a sharp bound on the number of triangles in a graph with fixed number of edges. We also characterize graphs that achieve the maximum number of triangles. Using the upper bound on number of triangles, we prove that if $G$ is a special $p$-group of rank $2 \leq k \leq \binom{ d}{ 2}$, then $|\mathcal{ M}(G)| \leq p^{ \frac{ d(d+2k-1)}{ 2} - k- \binom{ d}{ 3}+ \binom{ r}{ 3} + \mybinom[.55]{ \binom{ d}{ 2} - k - \binom{ r}{ 2} }{ 2} }$, where $r$ is such that $\binom{ r}{ 2} \leq \binom{ d}{ 2} -k < \binom{ r+1}{ 2} $. We also prove that, if $G$ is a $p$-group $(p \neq 2,3)$ of class $c \geq 3$, then $|\mathcal{ M}(G)| \leq p^{ \frac{ d(m-e)}{ 2}+(\delta-1)(n-m)-\max(0,\delta-2)-\max(1,\delta-3)}$ and if $G$ is of coclass $r$ with class $c \geq 3$, then $|\mathcal{ M}(G)| \leq p^{ \frac{ r^2-r}{ 2}+kr}$
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