Semicomplete multipartite weakly distance-regular digraphs
Shuang Li, Yuefeng Yang, Kaishun Wang
TL;DR
This work advances the theory of semicomplete multipartite weakly distance-regular digraphs by extending Jørgensen et al.’s doubly regular team tournaments to the broader class of doubly regular team semicomplete multipartite digraphs, proving a three-type classification. Using association schemes, quotient constructions, and detailed case analysis based on the two-way distance set $T$, the authors characterize all semicomplete multipartite commutative weakly distance-regular digraphs as explicit families built from coclique extensions and well-understood base digraphs (including Cayleylike structures on $\mathbb{Z}_6$ and products with cycles). The paper establishes necessary and sufficient conditions for the possible $T$-configurations ($T=\{3\},\{4\},\{3,4\},\{2,3\}$) and then derives complete classifications for each case, culminating in a proof of the main theorem. The results provide new, concrete classifications with potential implications for the study of non-symmetric association schemes and distance-regular-like properties in digraphs, aiding SEO through precise combinatorial constructions and nomenclature such as coclique extensions and doubly regular team tournaments.
Abstract
A digraph is semicomplete multipartite if its underlying graph is a complete multipartite graph. As a special case of semicomplete multipartite digraphs, Jørgensen et al. \cite{JG14} initiated the study of doubly regular team tournaments. As a natural extension, we introduce doubly regular team semicomplete multipartite digraphs and show that such digraphs fall into three types. Furthermore, we give a characterization of all semicomplete multipartite commutative weakly distance-regular digraphs.