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Exponents of Orientable Maps
Author(s) -
Nedela R,
Škoviera Martin
Publication year - 1997
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611597000245
Subject(s) - mathematics , exponent , combinatorics , automorphism , automorphism group , abelian group , permutation group , group (periodic table) , vertex (graph theory) , discrete mathematics , graph , permutation (music) , philosophy , linguistics , physics , chemistry , organic chemistry , acoustics
We generalize the idea of reflexibility of a map on a surface by introducing certain integers as its ”exponents“. An exponent is any integer e with the property that changing the cyclic permutation of edges around each vertex induced by the map to its e ‐th power gives rise to an isomorphic map. The exponents reduced modulo the least common multiple of the vertex valencies form an Abelian group, the exponent group Ex( M of the map M . Along with the automorphism group, the group in fact provides an additional measure of symmetry of ∼ M . The paper is devoted to developing the fundamentals of the theory of exponent groups of maps. Motivation comes from the problem of classification of regular maps with a given underlying graph. To this end, we prove that the number of non‐isomorphic regular maps (if any) with a given underlying graph and the same map automorphism group is |Z n ∗ : Ex ( M ) | , n being the valency of M . We calculate the exponent groups for some interesting families of regular maps including complete maps. Special attention is paid to the problem of how the antipodality of a map is reflected by its exponent group. In the final section we discuss several open problems. 1991 Mathematics Subject Classification : 05C10, 05C25, 20F32.