Originally Posted by

**jameselmore91** I'm taking Group theory next semester and have started doing some preliminary studying, the online book I found has practice problems in it and I'm having a hard time with this particular problem:

"The complete m-partite graph on n vertices in which each part has n/m vertices is denoted by $\displaystyle T_{m,n}$. Show that:

$\displaystyle \epsilon(T_{m,n}) = \left(\begin{array}{cc}n-k\\2\end{array}\right) + (m-1)\left(\begin{array}{cc}k+1\\2\end{array}\right)$ where k is $\displaystyle n/m$"

I hope someone has a solution!

Thank in advance,

James Elmore

To me it seems that the binomial coefficient stuff is just a ploy. I feel a simple argument shows that the total number of edges is (forgetting your definition of $\displaystyle k$, I'll just write $\displaystyle \frac{n}{m}$)

$\displaystyle \displaystyle \begin{aligned}\sum_{k=1}^{m-1}\frac{n}{m}\left(\frac{m-k}{m}n\right) &=\frac{n^2}{m^2}\sum_{k=1}^{m-1}\left(m-k\right)\\ &=\frac{n^2}{m^2}\left(m(m-1)+\frac{m(m-1)}{2}\right)\\ &= {{n-\frac{n}{m}}\choose 2}+(m-1){{\frac{n}{m}+1}\choose 2}\end{aligned}$

P.S. What the hell does this have to do with group theory?