论文标题
图中最大的三角形及其在特殊$ p $ groups的应用
The maximum number of triangles in a graph and its applications to special $p$-groups
论文作者
论文摘要
我们对图表中的三角形数量有一个锐利的结合，并具有固定数量的边缘数。我们还表征了达到最大三角形数量的图。使用三角形数量的上限，我们证明，如果$ g $是特殊的$ p $ - 等级$ 2 \ leq k \ leq \ binom {d} {2} {2} $，则$ | \ mathcal {m {m}（m}（g）| \ leq p^{\ frac {d（d+ 2k -1）} {2} - k- \ binom {d} {d} {3}+ \ binom {r} {3}+ \ \ \ \ \ \\ mybinom [.55] {\ binom {\ binom {d} } $，其中$ r $是如此，以至于$ \ binom {r} {2} \ leq \ binom {d} {2} {2} -k <\ binom {r+1} {2} {2} $。我们还证明，如果$ g $是$ p $ -group $（p \ neq 2,3）$ c \ geq 3 $，则是$ | \ mathcal {m}（g）| \ leq p^{\ frac {d（m-e）} {2}+（δ-1）（n-m） - \ max（0，δ-2） - \ max（1，δ-3）$，如果$ g $是coclass $ r $，带有class $ c \ geq 3 $，然后$ c \ geq 3 $，然后\ leq p^{\ frac {r^2-r} {2}+kr} $
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 $|\mathcal{M}(G)| \leq p^{\frac{d(m-e)}{2}+(δ-1)(n-m)-\max(0,δ-2)-\max(1,δ-3)}$ and if $G$ is of coclass $r$ with class $c \geq 3$, then $|\mathcal{M}(G)| \leq p^{\frac{r^2-r}{2}+kr}$
