A digraph version of the friendship theorem (Q6635106)

The Friendship Theorem states that if any pair of people in a group has precisely one friend in common, then there is always one person who is everyone's friend. This theorem was proved by \textit{P. Erdős} et al. [Stud. Sci. Math. Hung. 1, 215--235 (1966; Zbl 0144.23302)]. This relation is modeled in so-called friendship graphs.\N\NThe authors study the relation liking instead of friendship. Unlike friendship, this relation may not be symmetric. The corresponding liking graphs are digraphs. It is proved that if a digraph is a liking digraph, then it is a fancy wheel digraph or it is a \(k\)-diregular digraph of order \(k^2-k+1\) for some integer \(k \geq 2\). A fancy wheel digraph is a digraph that is obtained from the disjoint union of directed cycles by adding one vertex \(v\) and two arcs in both directions between \(v\) and any vertex of the cycles. It is also shown, that for each \(k \geq 3\), there exists a \(k\)-diregular liking digraph if and only if there exists a projective plane of order \(k-1\).