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arXiv · 2405.02662

Digraphs in which every $t$ vertices have exactly $\lambda$ common out-neighbors

Myungho Choi·Hojin Chu·Suh-Ryung Kim
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Abstract

We say that a digraph is a $(t,\lambda)$-liking digraph if every $t$ vertices have exactly $\lambda$ common out-neighbors. In 1975, Plesn\'{i}k [Graphs with a homogeneity, 1975. {\it Glasnik Mathematicki} 10:9-23] proved that any $(t,1)$-liking digraph is the complete digraph on $t+1$ vertices for each $t\geq 3$. Choi {\it et al}. [A digraph version of the Friendship Theorem, 2025. {\it Discrete mathematics}, 348(1), 114238] 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 $(t,\lambda)$-liking digraphs with $t \geq \lambda+2$ and giving some equivalent conditions for a $(t,\lambda)$-liking digraph being a complete digraph on $t+\lambda$ vertices.

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Myungho Choi, Hojin Chu, Suh-Ryung Kim. 2024-05-04. Digraphs in which every $t$ vertices have exactly $\lambda$ common out-neighbors. https://arxiv.org/abs/2405.02662

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