Publisher : Department of Mathematics, Kyungpook National University
DOI : 10.5666/KMJ.2014.54.4.655
Title & Authors
Enumerations of Finite Topologies Associated with a Finite Simple Graph Kim, Dongseok; Kwon, Young Soo; Lee, Jaeun;
The number of topologies (non-homeomorphic topologies) on a fixed finite set having n elements are now known up to n = 18 (n = 16 respectively) but still no complete formula yet. There are one to one correspondences among topologies, preorders and transitive digraphs on a given finite set. In this article, we enumerate topologies and non-homeomorphic topologies whose underlying graph is a given finite simple graph.
L. W. Brinn, Computing topologies, Mathematics Magazine, 58(1985), 67-77.
J. W. Evans, F. Harary and M. S. Lynn, On the computer enumeration of finite topologies, Communications of the ACM, 10(1967), 295-297.
J. Gross and T. Tucker, Topological graph theory, Wiley-Interscience Series in discrete Mathematics and Optimization, Wiley & Sons, New York, 1987.
V. A. Kovalevsky, Finite topology as applied to image analysis, Computer Vision, Graphics, and Image Processing, 46(1989), 141-161.
V. A. Kovalevsky, Finite topology and image analysis, Advances in electronics and electron physics, 84(1992), 197-324.
C. Marijuan, Finite topology and digraphs, Proyecciones Journal of Mathematics, 29(2010), 291-307.
R. Merrifield and H. Simmons, The structures of molecular topological spaces, Theoretica Chimica Acta, 55(1980), 55-75.
R. Merrifield and H. Simmons, Topological methods in chemistry, Wiley, New York, 1989.
K. Ragnarsson and B. E. Tenner, Obtainable sizes of topologies on finite sets, Journal of Combinatorial Theory, Series A, 117(2010), 138-151, arXiv:0802.2550
A. Rosenfeld, T. Kong and A. Wu, Digital surfaces, Graphical Models and Image Processing, 53(1991), 305-312.
N. J. A. Sloane, On-Line Encyclopedia of Integer Sequences, A000798, A001035, A001930, A000112. Available at http://oeis.org/.
J. R. Stallings, Topology of finite graphs, Invent. math., 71(1983), 551-565.
T. Van Zandt. PSTricks: PostScript macros for generic TEX. Available at ftp://ftp.princeton.edu/pub/tvz/.