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Weakly distance-regular digraphs. (English) Zbl 1014.05077
Summary: We consider the following generalization of distance-regular digraphs. A connected digraph \(\Gamma\) is said to be weakly distance-regular if, for all vertices \(x\) and \(y\) with \((\partial(x,y),\partial(y,x))=\widetilde h\), \(|\{z\in V\Gamma\mid (\partial(x, z),\partial(z,x))=\widetilde i\) and \((\partial(z,y),\partial(y,z))=\widetilde j\}|\) depends only on \(\widetilde h\), \(\widetilde i\) and \(\widetilde j\). We give some constructions of weakly distance-regular digraphs and discuss the connections to association schemes. Finally, we determine all commutative weakly distance-regular digraphs of valency 2.

MSC:
05E30 Association schemes, strongly regular graphs
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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