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arXiv · 2404.03910

On combinatorial structure and algebraic characterizations of distance-regular digraphs

Giusy Monzillo·Safet Penić
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Abstract

Let $\Gamma=\Gamma(A)$ denote a simple strongly connected digraph with vertex set $X$, diameter $D$, and let $\{A_0,A:=A_1,A_2,\ldots,A_D\}$ denote the set of distance-$i$ matrices of $\Gamma$. Let $\{R_i\}_{i=0}^D$ denote a partition of $X\times X$, where $R_i=\{(x,y)\in X\times X\mid (A_i)_{xy}=1\}$ $(0\le i\le D)$. The digraph $\Gamma$ is distance-regular if and only if $(X,\{R_i\}_{i=0}^D)$ is a commutative association scheme. In this paper, we describe the combinatorial structure of $\Gamma$ 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.

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Giusy Monzillo, Safet Penić. 2024-04-05. On combinatorial structure and algebraic characterizations of distance-regular digraphs. https://arxiv.org/abs/2404.03910

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