Maximal graphical realization of a topology
DOI:
https://doi.org/10.22199/issn.0717-6279-5696Keywords:
Set-indexer, topology, t-set-graceful, optimal spaceAbstract
Given a topological space, the graphical realizations of it with as many edges as possible, called maximal graphical realizations, are studied here. Every finite topological space admits a maximal graphical realization. However, there are graphs which are not maximal graphical realizations of any topology. A tree of odd order is never a maximal graphical realization of a topological space. Maximal graphical realization of a topology is a cycle if and only if it is C_3. It is shown that chain topologies admit unique maximal graphical realizations. A lower bound for the size of a maximal graphical realization is also obtained.
References
Acharya B D, Set valuations of a graph and their applications. MRI Lecture Notes in Applied Mathematics No.2, Mehta Research Institute, Allahabad, (1983).
Acharya B D, Set valuations of graphs and their applications. Proc. Sympos. on Optimization, Design of Experiments and Graph Theory I.I.T. Bombay (1986) 231-238.
Hegde S M, On set valuations of graphs, Nat. Acad. Sci. Letters 14(4) (1991) 181-182.
Hegde S M, On set colourings of graphs, Eur. J. Comb. 30(4) (2009) 986-995.
E. Cech, Topological Spaces (1966) Wiley.
Mathew S C and Thomas U, Strongly t-set graceful graphs, Graph Theory Notes N. Y. LXII (2012) 17-28.
Mollard M and Payan C, On two conjectures about set-graceful graphs, European J. Combin. 10 (1989) 185-187.
Princy K L, Some Studies on Set Valuation of Graphs-Embedding and NP Completeness, (2007) Ph. D. Thesis, Kannur University.
Shiu W C and Lam P C B, Super-edge-graceful labelings of multi-level wheel graphs, fan graphs and actinia graphs, Congr. Numer. 174 (2005) 49-63.
Thomas U and Mathew S C, On set-indexers of paths, cycles and certain related graphs, Discrete Math. Algorithms Appl. 4(3) (2012) 1250025 (12 pages).
Thomas U and Mathew S C, On topological set indexers of graphs, Adv. Appl. Discrete Math. 5(2) (2010) 115-130.
Thomas U and Mathew S C, Topologically set-graceful stars, paths and related graphs, South Asian J. Math. 2(4) (2012) 392-408.
Thomas U and Mathew S C, Graphical realization of a topology, J. Adv. Res. Pure Math. 5(3) (2013) 73-87.
Thomas U and Mathew S C, On Set-Indexers of Graphs, Palestine Journal of Mathematics 3(2)(2014) 273-280.
Thomas U and Mathew S C, On topological numbers of graphs, Novi Sad J. Math. 45(2) (2015) 85-95.
Stong R E, Finite topological spaces, Trans. Amer. Math. Soc. 123(2) (1966) 325-340.
Willard S, General Topology (1970) Addison-Wesley.
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