# zbMATH — the first resource for mathematics

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.)
Full Text: