A doubly regular tournament is a tournament such that every two vertices have $j$ common out-neighbours. How can we prove such a tournament is $2j+1$-regular?



How to prove that doubly regular tournaments are regular?
Let $v$ be any vertex, $U$ be a set of its out-neighbours. The restriction of your tournament to $U$ is $j$-out-regular, thus $|U|=2j+1$.


