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Examples to a theorem of Weiss. (English) Zbl 0833.05040

This paper constructs infinite families of graphs with the following properties: The graphs have a vertex-transitive group of automorphisms whose vertex-stabilizer subgroup induces on the neighbors a primitive group containing an abelian regular normal subgroup; furthermore the subgroup fixing a vertex and all its neighbors pointwise has non-trivial intersection with the corresponding subgroup for an adjacent vertex.
The referenced theorem of R. Weiss [Math. Proc. Cambridge Philos. Soc. 85, 43-48 (1979; Zbl 0392.20002)] proves that, given these conditions, the graph has degree (valency) either \(3^n\) or \(4^n\) for some \(n\) (and several conclusions can be drawn about the group). The author constructs examples of degree \(3^n\) and \(4^n\) for every \(n\) (previous examples had degree 3 or 4). The construction is simply a type of \(n\)-fold Cartesian product, where \(n\)-tuples are adjacent when all corresponding positions are adjacent (distinct) vertices of the original graph (only one connected component of the result is used). The resulting graph has degree \(d^n\) if the original graph has degree \(d\) (and its automorphism group is essentially the wreath product of the original automorphism group with \(\text{Sym}_n\)).
Four well-known graphs are listed (two of degree 3 and two of degree 4) whose \(n\)-fold products produce the infinite families with all the desired properties. Finally the author mentions further examples for a possible generalization of the theorem of Weiss.

MSC:

05C25 Graphs and abstract algebra (groups, rings, fields, etc.)

Citations:

Zbl 0392.20002
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References:

[1] C.Berge, Teorié des graphes et ses applications. Paris 1958.
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[3] G. F.Picard, Latticoid products, sums and products of graphs. In: Combinatorial structures and their applications, 321-322. New York 1970.
[4] G. F. Picard, Distributivité d’opérations sur les graphes. C. R. Acad. Sci. Paris Sér. I Math.270, 1219-1221 (1970). · Zbl 0194.56203
[5] R. Weiss, An application ofp-factorization methods to symmetric graphs. Math. Proc. Cambridge Philos. Soc.85, 43-48 (1979). · Zbl 0392.20002 · doi:10.1017/S030500410005547X
[6] R. Weiss, Groups with an (B, N)-pair and locally transitive graphs. Nagoya Math. J.74, 1-21 (1979). · Zbl 0381.20004
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[8] R.Weiss, Personal communication, 1993.
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