Weakly distance-regular digraphs of one type of arcs

TL;DR

This paper classifies a specific class of weakly distance-regular directed graphs with one arc type, girth g, under certain algebraic conditions, extending previous results in the field.

Contribution

It provides a complete classification of commutative weakly distance-regular digraphs of girth g with one arc type under a key algebraic inequality.

Findings

Classification of all such digraphs under the given conditions

Recovery of a known theorem as a special case

Extension of previous classification results

Abstract

In this paper, we classify all commutative weakly distance-regular digraphs of girth $g$ and one type of arcs under the assumption that $p_{(1,g-1),(1,g-1)}^{(2,g-2)}\geq k_{1,g-1}-2$. In consequence, we recover [13, Theorem 1.1] as a special case of our result.