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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2404.03910 |
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| _version_ | 1866909161897328640 |
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| author | Monzillo, Giusy Penić, Safet |
| author_facet | Monzillo, Giusy Penić, Safet |
| contents | Let $Γ=Γ(A)$ 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 $Γ$. 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 $Γ$ 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 $Γ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_03910 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On combinatorial structure and algebraic characterizations of distance-regular digraphs Monzillo, Giusy Penić, Safet Combinatorics Let $Γ=Γ(A)$ 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 $Γ$. 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 $Γ$ 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 $Γ$ 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. |
| title | On combinatorial structure and algebraic characterizations of distance-regular digraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2404.03910 |