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why must a tounrament on n vertices only have one vertex of outdegree n-1?

why must a tounrament on n vertices only have one vertex of outdegree n-1?
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Answer #1

In a tournament on   n vertices, there is at most one vertex of outward degree n-1 . In some case there cannot exist a vertex of degree n-1 . for example when n=3 , the graph

1---->2---->3---->1, has no vertex of degree 2.

But if one exists, it is unique. If 'v 'is vertex of out degree 'n-1', then every edge is going out from 'v'. That implies, for every other vertex 'w' the edge joining 'v' and 'w' is coming into 'w'. Hence, the number of edges going out from 'w' is at most 'n-2'(edges connecting 'w' and other n-2 vertices with 'w' and 'v' excluded).

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