[Paper Review] Quasi-thin weakly distance-regular digraphs

Yuefeng Yang, Benjian Lv+1 more|arXiv (Cornell University)|Sep 16, 2016

Finite Group Theory Research8 references3 citations

TL;DR

This paper classifies commutative quasi-thin weakly distance-regular digraphs with valency greater than 3, proving they are isomorphic to specific families of Cayley digraphs over abelian groups. The classification relies on intersection number analysis and association scheme theory, yielding a complete structural characterization of these digraphs.

ABSTRACT

A weakly distance-regular digraph is quasi-thin if the maximum value of its intersection numbers is 2. In this paper, we focus on commutative quasi-thin weakly distance-regular digraphs, and classify such digraphs with valency more than 3. As a result, this family of digraphs are completely determined.

Motivation & Objective

• To classify commutative quasi-thin weakly distance-regular digraphs with valency exceeding 3.
• To determine the complete structural family of such digraphs using algebraic and combinatorial techniques.
• To establish that all such digraphs are isomorphic to specific Cayley digraphs over abelian groups.
• To extend prior classifications of weakly distance-regular digraphs to the quasi-thin, commutative case with higher valency.
• To provide a complete and explicit list of all such digraphs via group-theoretic constructions.

Proposed method

• Utilizes the framework of association schemes derived from weakly distance-regular digraphs, with adjacency relations indexed by two-way distances.
• Applies intersection number identities (e.g., $ p_{ ilde{d}, ilde{e}}^{ ilde{f}} k_{ ilde{f}} = p_{ ilde{f}, ilde{e}^*}^{ ilde{d}} k_{ ilde{d}} $) to constrain possible configurations.
• Employs group-theoretic constructions, particularly Cayley digraphs over $ bZ_n imes bZ_m $, to generate candidate families.
• Uses isomorphism theorems and partitioning of vertex sets into distance-level sets to verify structural consistency.
• Applies case analysis based on intersection number values (e.g., $ k_{1,1} = 1 $ or $ 2 $) to distinguish between different families.
• Verifies isomorphism via explicit mappings (e.g., $ au, ho, heta $) between constructed digraphs and the canonical Cayley forms.

Experimental results

Research questions

• RQ1 Which commutative quasi-thin weakly distance-regular digraphs exist with valency greater than 3?
• RQ2 What are the necessary and sufficient conditions for a Cayley digraph to be weakly distance-regular and quasi-thin?
• RQ3 How do intersection numbers and two-way distances constrain the global structure of such digraphs?
• RQ4 Can all such digraphs be completely classified into finitely many families with explicit group and connection set descriptions?
• RQ5 What role do group orders and gcd conditions (e.g., $ ext{gcd}(q,n) $) play in determining valid connection sets?

Key findings

• All commutative quasi-thin weakly distance-regular digraphs with valency greater than 3 are isomorphic to one of ten explicitly listed Cayley digraphs.
• The classification includes families over $ bZ_8 $, $ bZ_{4p} $, $ bZ_4 imes bZ_4 $, $ bZ_q imes bZ_4 $, and products with $ bZ_n $, each with specific connection sets.
• For digraphs with $ k_{1,1} = 1 $, the structure is isomorphic to $ ext{Cay}(bZ_{4q}, igracevert ext{Cay}(bZ_{4q}, igracevert $, with $ q e 3 $.
• When $ k_{1,1} = 2 $, the digraphs are isomorphic to $ ext{Cay}(bZ_{2q} imes bZ_n, igracevert $, with $ n e 3 $, and $ c = n / ext{gcd}(q,n) $, $ t = q / ext{gcd}(q,n) $ both odd.
• The proof establishes that no other such digraphs exist beyond the ten families listed in Theorem 1.1.
• The isomorphisms are explicitly constructed via mappings like $ heta(a,b,i) = (2 ilde{a}+i, (2 ilde{a}c + ic + i)/2 + ilde{b}) $, confirming structural equivalence.