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

In this paper, we introduce the notion of two-way (t, ) -liking digraphs as a way to extend the results for generalized friendship graphs. A two-way (t, ) -liking digraph is a digraph in which every t vertices have exactly common out-neighbors and common in-neighbors. We first show that if 2, then a two-way (2, ) -liking digraph of order n is k-diregular for a positive integer k satisfying the equation (n-1) =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+ vertices is the only two-way (t, ) -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, ) -liking digraphs are closely linked to symmetric block designs and extend some existing results of (t, ) -liking digraphs.