Abstract
Lifts of graph and map automorphisms can be described in terms of voltage assignments that are, in a sense, compatible with the automorphisms. We show that compatibility of ordinary voltage assignments in Abelian groups is related to orthogonality in certain \(\mathcal{Z}\)-modules. For cyclic groups, compatibility turns out to be equivalent with the existence of eigenvectors of certain matrices that are naturally associated with graph automorphisms. This allows for a great simplification in characterizing compatible voltage assignments and has applications in constructions of highly symmetric graphs and maps.
Article PDF
Similar content being viewed by others
References
D. Archdeacon, “The medial graph and voltage-current duality,” Discrete Math. 104 (1992), 111–141.
D. Archdeacon, B. Richter, J. Širáň, and M. Škoviera, “Branched coverings of maps and lifts of map homomorphisms,” Australas. J. Combinat. 9 (1994), 109–121.
D. Archdeacon, J. Širáň, and M. Škoviera, “Self-dual regular maps from medial graphs,” Acta Math. Univ. Comenianae LXI (1992), 57–64.
N.L. Biggs, “Homological coverings of graphs,” J. London Math. Soc. 30 (1984), 1–14.
J.A. Bondy and U.S.R. Murty, Graph Theory with Applications, American Elsevier, New York, 1976.
D.Ž. Djokovic, “Automorphisms of graphs and coverings,” J. Combinat. Theory Ser. B 16 (1974), 243–247.
J.L. Gross and T.W. Tucker, Topological Graph Theory, Wiley-Interscience, New York, 1987.
P. Gvozdjak and J. Širáň, “Regular maps from voltage assignments,” in Graph Structure Theory, N. Robertson and P. Seymour (Eds.), Contemporary Mathematics (AMS Series), Vol. 147, 1993, pp. 441–454.
G.A. Jones, “Maps on surfaces and Galois groups,” Math. Slovaca 47 (1997), 1–33.
G.A. Jones and D. Singerman, “Theory of maps on orientable surfaces,” Proc. London Math. Soc. 37(3) (1978), 273–307.
G.A. Jones and D. Singerman, “Belyi functions, hypermaps, and Galois groups,” Bull. London Math. Soc. 28 (1996), 561–590.
J.H. Kwak and J. Lee, “Isomorphism classes of Mgraph bundles,” Canad. J. Math. XLII (1990), 747–761.
A. Malnič, R. Nedela, and M. Škoviera, “Lifting graph automorphisms by voltage assignments,” Europ. J. Combin. 21 (2000), 927–947.
W.S. Massey, Algebraic Topology: An Introduction, Harcourt Brace and World, New York, 1967.
R.B. Richter, J. Širáň, R. Jajcay, T.W. Tucker, and M.E. Watkins, “Cayley maps,” submitted.
R.B. Richter and W.P. Wardlaw, “Diagonalization over commutative rings,” Amer. Math. Monthly 97 (1990), 223–227.
D.B. Surowski, “Lifting map automorphisms and MacBeath's theorem,” J. Combinat. Theory Ser. B 50 (1988), 135–149.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Širáň, J. Coverings of Graphs and Maps, Orthogonality, and Eigenvectors. Journal of Algebraic Combinatorics 14, 57–72 (2001). https://doi.org/10.1023/A:1011218020755
Issue Date:
DOI: https://doi.org/10.1023/A:1011218020755