Abstract
We study the problem of determining when two Cayley graphs on a given abelian group are isomorphic. The emphasis is on abelian p-groups.
This work was partially supported by the Natural Sciences and Engineering Research Council of Canada under Grant A-4792. The author wishes to thank Aubert Daigneault, Geňa Hahn, Gert Sabidussi and the Department of Mathematics of the Université de Montréal for their support and organizational skills.
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References
A. Adam, Research Problem 2–10, J. Combin. Theory 2 (1967), 393.
B. Alspach, Point-symmetric graphs and digraphs of prime order and transitive permutation groups of degree prime degree, J. Combin. Theory 15 (1973), 12–17.
L. Babai, Isomorphism problem for a class of point-symmetric structures, Acta Math. Acad. Sci. Hungar. 29 (1977), 329–336.
L. Babai and P. Frankl, Isomorphisms of Cayley graphs I, in: Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely 1976), Colloq. Math. Soc. Janos Bolyai 18, North-Holland, Amsterdam, 1978, 35–52.
J. L. Berggren, An algebraic characterization of symmetric graphs with p points, p and odd prime, Bull. Austral. Math. Soc. 7 (1972), 131–134.
F. Boesch and R. Tindell, Circulants and their connectivities, J. Graph Theory 8 (1984), 487–499.
N. G. de Bruijn, Pólya’s theory of counting, in: Applied Combinatorial Mathematics, Wiley, New York, 1964.
C. Delorme, O. Favaron and M. Maheo, Isomorphisms of Cayley multigraphs of degree 4 on finite abelian groups, European J. Combin. 13 (1992), 59–61.
E. Dobson, Isomorphism problem for Cayley graphs of Zp, Discrete Math. 147 (1995), 87–94.
B. Elspas and J. Turner, Graphs with circulant adjacency matrices, J. Combin. Theory 9 (1970), 297–307.
X.-G. Fan and M.-Y. Xu, On isomorphisms of Cayley graphs of small valency, Algebra Colloq. 1 (1994), 67–76.
R. Frucht, Herstellung von Graphen mit vorgegebener abstrakter Gruppe, Compositio Math. 6 (1938), 239–250.
R. Frucht, Graphs of degree three with a given abstract group, Canad. J. Math. 1 (1949), 365–378.
C. D. Godsil, On Cayley graph isomorphisms, Ars Combin. 15 (1983), 231–246.
H. Izbicki, Unendliche Graphen endlichen Grades mit vorgegebenen Eigenschaften, Monatsh. Math. 63 (1959), 298–301.
A. Joseph, The isomorphism problem for Cayley digraphs on groups of prime-squared order, Discrete Math. 141 (1995), 173–183.
D. König, Theorie der endlichen und unendlichen Graphen, Akad. Verlagsgesellschaft, Leipzig 1936; reprinted by Chelsea, New York, 1950.
C.-H. Li, Isomorphisms and classification of Cayley graphs of small valencies on finite abelian groups, Austral. J. Combin. 12 (1995), 3–14.
J. Lützen, G. Sabidussi and B. Toil, Julius Petersen 1839–1910: A biography, Discrete Math. 100 (1992), 9–82.
J. Morris, Isomorphic Cayley graphs on nonisomorphic groups, preprint, Simon Fraser University 1996.
M. Muzychuk, Adam’s conjecture is true in the square-free case, J. Combin. Theory Ser. A 72 (1995), 118–134.
M. Muzychuk, On Adam’s conjecture for circulant graphs, Discrete Math.,to appear.
L. Nowitz, A non-Cayley-invariant Cayley graph of the elementary Abelian group of order 64, Discrete Math. 110 (1992), 223–228.
P. P. Palfy, Isomorphism problem for relational structures with a cyclic automorphism, European J. Combin. 8 (1987), 35–43.
D. S. Passman, Permutation Groups, Benjamin, Menlo Park, CA, 1968.
G. Sabidussi, Graphs with given group and given graph-theoretical properties, Canad. J. Math. 9 (1957), 515–525.
S. Toida, A note on Adam’s conjecture, J. Combin. Theory Ser. B 23 (1977), 239–246.
J. Turner, Point-symmetric graphs with a prime number of points, J. Combin. Theory 3 (1967), 136–145.
H. Wielandt, Finite Permutation Groups, Academic Press, New York, 1964.
M.-Y. Xu, On isomorphisms of Cayley digraphs and graphs of groups of order p3, Adv. in Math. (China) 17 (1988), 427–428.
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Alspach, B. (1997). Isomorphism and Cayley graphs on abelian groups. In: Hahn, G., Sabidussi, G. (eds) Graph Symmetry. NATO ASI Series, vol 497. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8937-6_1
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DOI: https://doi.org/10.1007/978-94-015-8937-6_1
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