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

TL;DR

This paper introduces two-way $(t, ext{lambda})$-liking digraphs, extending classical results by characterizing their structure, regularity, and uniqueness, and linking them to symmetric block designs.

Contribution

It defines two-way $(t, ext{lambda})$-liking digraphs and extends known results on their structure, regularity, and uniqueness, connecting them to symmetric block designs.

Findings

For $ ext{lambda} extgreater= 2$, such digraphs are $k$-diregular with $(n-1) ext{lambda}=k(k-1)$.

Complete digraphs on $t+ ext{lambda}$ vertices are the only two-way $(t, ext{lambda})$-liking digraphs for $t extgreater= 3$.

These digraphs are closely linked to symmetric block designs.

Abstract

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