Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

An infinite graph of girth $12$
HTML articles powered by AMS MathViewer

by Asia Ivić Weiss PDF
Trans. Amer. Math. Soc. 283 (1984), 575-588 Request permission

Abstract:

From the regular hyperbolic honeycomb $\{ 3,6,3\}$ we derive regular honeycombs with finite numbers of toroidal cells. Joining centers of faces of these honeycombs to the midpoints of its edges we obtain trivalent symmetrical graphs. We investigate the relation between these honeycombs, their groups and the graphs embedded in them.
References
  • Luigi Bianchi, Geometrische Darstellung der Gruppen linearer Substitutionen mit ganzen complexen Coefficienten nebst Anwendungen auf die Zahlentheorie, Math. Ann. 38 (1891), no. 3, 313–333 (German). MR 1510677, DOI 10.1007/BF01199425
  • N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
  • H. S. M. Coxeter, The abstract groups $G^{m,n,p}$, Trans. Amer. Math. Soc. 45 (1939), no. 1, 73–150. MR 1501984, DOI 10.1090/S0002-9947-1939-1501984-9
  • H. S. M. Coxeter, Twelve geometric essays, Southern Illinois University Press, Carbondale, Ill.; Feffer & Simons, Inc., London-Amsterdam, 1968. MR 0310745
  • H. S. M. Coxeter, Regular polytopes, 3rd ed., Dover Publications, Inc., New York, 1973. MR 0370327
  • H. S. M. Coxeter, Regular complex polytopes, Cambridge University Press, London-New York, 1974. MR 0370328
  • H. S. M. Coxeter, The edges and faces of a $4$-dimensional polytope, Congr. Numer. 28 (1980), 309–334. MR 608436
  • Harold Scott Macdonald Coxeter, Roberto Frucht, and David L. Powers, Zero-symmetric graphs, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. Trivalent graphical regular representations of groups. MR 658666
  • H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, 4th ed., Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 14, Springer-Verlag, Berlin-New York, 1980. MR 562913
  • H. S. M. Coxeter and G. J. Whitrow, World-structure and non-Euclidean honeycombs, Proc. Roy. Soc. London Ser. A 201 (1950), 417–437. MR 41576, DOI 10.1098/rspa.1950.0070
    • R. M. Foster, A census of trivalent symmetrical graphs. I, presented at the Conference on Graph Theory and Combinatorial Analysis, University of Waterloo, Ontario. C. W. L. Garner, Polyhedra and honeycombs in hyperbolic space, Ph.D. Thesis, University of Toronto. B. Grünbaum, Regularity of graphs, complexes and designs, Colloq. Internat. CNRS, Problèmes Combinatoires et Théorie des Graphes, Orsay. D. M. Y. Sommerville, An introduction to the geometry of $n$ dimensions, Methuen, London.
  • W. T. Tutte, Connectivity in graphs, Mathematical Expositions, No. 15, University of Toronto Press, Toronto, Ont.; Oxford University Press, London, 1966. MR 0210617
Similar Articles
Additional Information
  • © Copyright 1984 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 283 (1984), 575-588
  • MSC: Primary 52A45; Secondary 05C25, 20F05, 51M20
  • DOI: https://doi.org/10.1090/S0002-9947-1984-0737885-0
  • MathSciNet review: 737885