1. ## Graph Theory

Let T be a tournament on n vertices. Prove that:

A question from my book I am not sure how to do, anyone can show me how to solve or any tips on tackling this problem?

Thanks =)

Edit: all my stats were deleted :P

2. With a tournament of n vertices each vertex $\displaystyle v$ has:
$\displaystyle indeg(v) + outdeg(v) = n - 1$

$\displaystyle \sum (outdeg(v)^2 - indeg(v)^2)$

3. Originally Posted by snowtea
With a tournament of n vertices each vertex $\displaystyle v$ has:
$\displaystyle indeg(v) + outdeg(v) = n - 1$

$\displaystyle \sum (outdeg(v)^2 - indeg(v)^2)$

So the degree has to be n-1 because in a tournament every vertex is connected to every other vertex.

I'm having trouble wraping my head around the concept here... This is what I am understanding, in the tournament, the sum of all out degree must equal the sum of all in degree(intuitively for every out, there must be an in)

Maybe I am misunderstanding the question but what is the signifigance of squaring it? If x=y then of course $\displaystyle x^2=y^2$

$\displaystyle \sum (outdeg(v)^2 - indeg(v)^2)$ Would this not equal zero?

I'm sorry but I really do appreciate the help.

Thanks

4. Originally Posted by Len
$\displaystyle \sum (outdeg(v)^2 - indeg(v)^2)$ Would this not equal zero?