Digraphs in which every t vertices share exactly λ out-neighbors and exactly λ in-neighbors

In this paper, we introduce the notion of two-way (t,λ)-liking digraphs as a way to extend the results for generalized friendship graphs. A two-way (t,λ)-liking digraph is a digraph in which every t vertices have exactly λ common out-neighbors and λ common in-neighbors. We first show that if λ ≥ 2, then a two-way (2,λ)-liking digraph of order n is k-diregular for a positive integer k satisfying the equation (n-1)λ=k(k-1). This result is comparable to the result by Bose and Shrikhande in 1969 and actually extends it. Another main result is that if t ≥ 3, then the complete digraph on t+λ vertices is the only two-way (t,λ)-liking digraph. This result can stand up to the result by Carstens and Kruse in 1977 and essentially extends it. In addition, we find that two-way (t, λ)-liking digraphs are closely linked to symmetric block designs and extend some existing results of (t, λ)-liking digraphs.