The automorphism group of a product of graphs
Author:
Donald J. Miller
Journal:
Proc. Amer. Math. Soc. 25 (1970), 24-28
MSC:
Primary 05.62
DOI:
https://doi.org/10.1090/S0002-9939-1970-0262116-3
MathSciNet review:
0262116
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Abstract | References | Similar Articles | Additional Information
Abstract: In a recent paper we showed that every connected graph can be written as a weak cartesian product of a family of indecomposable rooted graphs and that this decomposition is unique to within isomorphisms. Using this unique prime factorization theorem we prove that if a graph $X$ can be written as a product of connected rooted graphs, which are pairwise relatively prime, then the automorphism group of $X$ is isomorphic to the restricted direct product of the automorphism groups of the factors with prescribed subgroups the isotropy groups of the factors at the roots. This is a generalization of Sabidussi’s theorem for cartesian multiplication.
- Wilfried Imrich, Über das schwache Kartesische Produkt von Graphen, J. Combinatorial Theory Ser. B 11 (1971), 1–16 (German, with English summary). MR 280401, DOI https://doi.org/10.1016/0095-8956%2871%2990008-6
- A. G. Kurosh, The theory of groups, Chelsea Publishing Co., New York, 1960. Translated from the Russian and edited by K. A. Hirsch. 2nd English ed. 2 volumes. MR 0109842
- Donald J. Miller, Weak cartesian product of graphs, Colloq. Math. 21 (1970), 55–74. MR 274327, DOI https://doi.org/10.4064/cm-21-1-55-74
- Gert Sabidussi, Graph multiplication, Math. Z. 72 (1959/60), 446–457. MR 209177, DOI https://doi.org/10.1007/BF01162967
- Gert Sabidussi, Vertex-transitive graphs, Monatsh. Math. 68 (1964), 426–438. MR 175815, DOI https://doi.org/10.1007/BF01304186
- N. Vilenkin, On direct decompositions of topological groups, Rec. Math. [Mat. Sbornik] N. S. 19(61) (1946), 85–154 (Russian, with English summary). MR 0017283
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Additional Information
Keywords:
Automorphism group,
weak cartesian product of graphs,
cartesian product of graphs,
restricted direct product of groups,
sum of graphs
Article copyright:
© Copyright 1970
American Mathematical Society