$P$-polynomial weakly distance-regular digraphs

TL;DR

This paper characterizes all $P$-polynomial weakly distance-regular digraphs, providing a complete classification based on their algebraic scheme properties.

Contribution

It offers a full characterization of $P$-polynomial weakly distance-regular digraphs, advancing the understanding of their algebraic and combinatorial structure.

Findings

Complete classification of $P$-polynomial weakly distance-regular digraphs

Identification of algebraic scheme properties defining these digraphs

Extension of the theory of distance-regular graphs to directed graphs

Abstract

A weakly distance-regular digraph is $P$-polynomial if its attached scheme is $P$-polynomial. In this paper, we characterize all $P$-polynomial weakly distance-regular digraphs.