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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

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: 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.


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DOI: https://doi.org/10.1090/S0002-9939-1970-0262116-3
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