English

On combinatorial structure and algebraic characterizations of distance-regular digraphs

Combinatorics 2024-04-08 v1

Abstract

Let Γ=Γ(A)\Gamma=\Gamma(A) denote a simple strongly connected digraph with vertex set XX, diameter DD, and let {A0,A:=A1,A2,,AD}\{A_0,A:=A_1,A_2,\ldots,A_D\} denote the set of distance-ii matrices of Γ\Gamma. Let {Ri}i=0D\{R_i\}_{i=0}^D denote a partition of X×XX\times X, where Ri={(x,y)X×X(Ai)xy=1}R_i=\{(x,y)\in X\times X\mid (A_i)_{xy}=1\} (0iD)(0\le i\le D). The digraph Γ\Gamma is distance-regular if and only if (X,{Ri}i=0D)(X,\{R_i\}_{i=0}^D) is a commutative association scheme. In this paper, we describe the combinatorial structure of Γ\Gamma in the sense of equitable partition, and from it we derive several new algebraic characterizations of such a graph, including the spectral excess theorem for distance-regular digraph. Along the way, we also rediscover all well-known algebraic characterizations of such graphs.

Keywords

Cite

@article{arxiv.2404.03910,
  title  = {On combinatorial structure and algebraic characterizations of distance-regular digraphs},
  author = {Giusy Monzillo and Safet Penić},
  journal= {arXiv preprint arXiv:2404.03910},
  year   = {2024}
}

Comments

arXiv admin note: text overlap with arXiv:2403.00652

R2 v1 2026-06-28T15:44:51.120Z