Semicomplete multipartite weakly distance-regular digraphs

TL;DR

This paper introduces and characterizes doubly regular team semicomplete multipartite digraphs, extending the study of weakly distance-regular digraphs within the framework of semicomplete multipartite graphs.

Contribution

It defines doubly regular team semicomplete multipartite digraphs, classifies their types, and provides a complete characterization of semicomplete multipartite commutative weakly distance-regular digraphs.

Findings

Doubly regular team semicomplete multipartite digraphs fall into three types.

Complete characterization of semicomplete multipartite commutative weakly distance-regular digraphs.

Extension of previous work on 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, J{\o}rgensen 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.