Question

A tournament is a simple directed graph such that if u and v are distinct vertices in the graph, exactly one of (u, v) and (v, u) is an edge of the graph.What is the sum of the in-degree and out-degree of a vertex in a tournament?

Short answer

The total sum of thein-degree and out-degree of a vertex is \(n - 1\).

Step-by-step solution

Concept/Significance of degree of a graph

The number of edges occurring on a vertex determines its degree in a directed graph.

Determination of the sum of the in-degree and out-degree of vertex

The tournament is a simple directed graph in which one of \(\left( {u,\;v} \right)\) and \(\left( {v,\;u} \right)\)is an edge of the graph if u and vare different vertices in the graph.

Then, the number of vertices in the tournament will be n andthere are still \(n - 1\) edges including vertex v because there is precisely one edge with endpoints \(u\)and \(v\), for each \(u\)differing from a given vertex \(v\).

As a result, the total in-degree and out-degree is \(n - 1\).

Thus, the total sum of thein-degree and out-degree of a vertex is\(n - 1\).