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Edge-symmetric distance-regular covers of cliques: affine case. (English. Russian original) Zbl 1270.05037
Dokl. Math. 87, No. 2, 224-228 (2013); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk, Vol. 449, No. 6, 639-643 (2013).
This paper studies edge-symmetric antipodal distance-regular graphs of diameter 3, based on a classification of twice transitive permutation groups. A connected graph $$G$$ is distance-regular if for any $$u$$ and $$v$$ in $$V(G)$$ and any integers $$i, j$$ in the range of $$[0, D(G)],$$ where $$D(G)$$ is the diameter of $$G,$$ the number of vertices at distance $$i$$ from $$u$$ and distance $$j$$ from $$v$$ depends only on $$i, j,$$ and the distance between $$u$$ and $$v,$$ but not on choices of $$u$$ and $$v$$ themselves. Such a graph can be characterized with its intersection array. It is also well known that a graph is edge symmetric if its automorphism group acts transitively on the set of its arcs (ordered edges). The whole paper contains one theorem and its proof. The statement of the theorem itself, together with clarification of all the involved terms, would easily fit in a page. Readers who are interested in this result should consult the paper itself.
This paper also contains a corollary of the above theorem, when the involved graph is vertex symmetric and the action of the group on the set of antipodal classes of the graph is affine.

MSC:
 05C12 Distance in graphs 05C60 Isomorphism problems in graph theory (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.) 05E18 Group actions on combinatorial structures
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References:
 [1] Godsil, C D; Liebler, R A; Praeger, C E, No article title, Eur. J. Comb., 19, 455-478, (1998) · Zbl 0914.05035 [2] A. E. Brouwer, A. M. Cohen, and A. Neumaier, Distance-Regular Graphs (Springer-Verlag, Berlin, 1989). · Zbl 0747.05073 [3] Makhnev, A A; Paduchikh, D V; Tsiovkina, L Yu, No article title, Dokl. Math., 87, 15-19, (2012) · Zbl 1273.05055 [4] Gavrilyuk, A L; Makhnev, A A, No article title, Dokl. Math., 78, 743-745, (2008) · Zbl 1250.05038 [5] W. Jones and B. Parshall, Proceedings of the Conference on Finite Groups (Academic, New York, 1976), pp. 313-327.
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