Digraphs in which every t vertices have exactly λ common out-neighbors

We say that a digraph is a (t,λ)-liking digraph if every t vertices have exactly λ common out-neighbors. In 1975, Plesn\'{i}k [Graphs with a homogeneity, 1975. { Glasnik Mathematicki} 10:9-23] proved that any $(t,1)$-liking digraph is the complete digraph on $t+1$ vertices for each t≥ 3. Choi { et al}. [A digraph version of the Friendship Theorem, 2023. arXiv preprint arXiv:2305.04058] (to appear in { Discrete mathematics}) showed that a $(2,1)$-liking digraph is a fancy wheel digraph or a k-diregular digraph for some positive integer k. In this paper, we extend these results by completely characterizing the (t,λ)-liking digraphs with t ≥ λ+2 and giving some equivalent conditions for a (t,λ)-liking digraph being a complete digraph on t+λ vertices.