Discrete Mathematics with Graph Theory (3rd Edition) 313

Discrete Mathematics with Graph Theory (3rd Edition) 313 -...

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Section 11.5 311 8. (a) The sum of all scores is the number of arcs (by Proposition 11.2.2) and in a tournament, this number is (~) = ~n(n - 1) because every two vertices are connected by a single arc. (b) Any t vertices VI, ... , Vt determine a digraph which is IC t with an arrow on each arc. The number of arcs in this digraph is m = ~ t( t - 1). So, by Proposition 11.2.2, the sums of the scores of the vertices VI, ... ,Vt just counting games between these players is ~t(t - 1). But the score of each Vi (in the entire tournament) is at least its score in the smaller digraph, so E Si :::: ~t(t - 1). 9. (a) Suppose V and w are vertices in a transitive tournament T. Either vw or wv must be an arc so, without loss of generality, we assume vw is an arc. If x is any vertex such that wx is an arc, so also is vx an arc, by transitivity. Remembering that vw is an arc, it follows that S ( v) :::: S ( w) + 1. In particular,
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This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.

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