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Homogeneous graphs and regular near polygons. (English) Zbl 0793.05130
A homogeneous graph $$\Gamma$$ is defined: for every edge $$uv$$, and vertex $$x$$, the number of edges from $$x$$ to $$\Gamma_ i(u)\cap\Gamma_ j(v)$$ depends only on $$i$$, $$j$$ and the distances from $$x$$ to $$u$$ and $$v$$. ($$\Gamma_ i(u)$$ is the set of vertices of distance $$i$$ from $$u$$.) It is proven that, for distance-regular graphs in which the set of common neighbors of adjacent vertices is always a clique, homogeneous graphs are precisely the regular near $$2d$$-gons.
Reviewer: K.Nomura

##### MSC:
 05C99 Graph theory 05C12 Distance in graphs 05E30 Association schemes, strongly regular graphs
##### Keywords:
homogeneous graph; distances; distance-regular graphs; clique
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