The maximum number of triangles in a graph and its applications to special $p$-groups

Abstract

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 $2\le k\le \left(\genfrac{}{}{0ex}{}{d}{2}\right)$, 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 $\left(\genfrac{}{}{0ex}{}{r}{2}\right)\le \left(\genfrac{}{}{0ex}{}{d}{2}\right)-k<\left(\genfrac{}{}{0ex}{}{r+1}{2}\right)$. We also prove that, if $G$ is a $p$-group $(p\ne 2,3)$ of class $c\ge 3$, then $|\mathcal{M}(G)|\le p^{\frac{d(m-e)}{2}+(\delta -1)(n-m)-\max (0,\delta -2)-\max (1,\delta -3)}$ and if $G$ is of coclass $r$ with class $c\ge 3$, then $|\mathcal{M}(G)|\le p^{\frac{r^2-r}{2}+kr}$