Abstract

Suzuki (2004) classified thin weakly distance-regular digraphs and proposed the project to classify weakly distance-regular digraphs of valency 3. The case of girth 2 was classified by the third author (2004) under the assumption of the commutativity. In this paper, we continue this project and classify these digraphs with girth more than 2 and two types of arcs.

Highlights

  • A digraph Γ is a pair (X, A) where X is a finite set of vertices and A ⊆ X2 is a set of arcs

  • An arc (u, v) of Γ is of type (1, r) if ∂(v, u) = r

  • A circuit is undirected if each of its arcs is of type

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Summary

Introduction

A digraph Γ is a pair (X, A) where X is a finite set of vertices and A ⊆ X2 is a set of arcs. A strongly connected digraph Γ is said to be distance-transitive if, for any vertices x, y, x and y of Γ satisfying. A strongly connected digraph Γ is said to be weakly distance-transitive if, for any vertices x, y, x and y satisfying ∂(x, y) = ∂(x , y ), there exists an automorphism σ of Γ such that x = σ(x) and y = σ(y). A strongly connected digraph Γ is said to be weakly distance-regular if, for all h, i, j. In [9], Wang classified all commutative weakly distance-regular digraphs of valency 3 and girth 2. Let Γ be a weakly distance-regular digraph of valency 3 and girth more than 2.

Constructions
Circuits
The proof of Theorem 1
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