A review on taxonomy of fuzzy graph
DOI:
https://doi.org/10.11113/mjfas.v13n1.650Abstract
Fuzzy graph is a graph that incorporates fuzziness. Fuzzy graph originated from the concept of Fuzzy Set and Graph. In this paper, taxonomy of fuzzy graph is reviewed. Several results on fuzzy graph are presented together with some examples. The fuzzy graph introduced by Yeh and Bang is proven to be a special case of Rosenfeld’s fuzzy graph. Two descriptions of isomorphic crisp graph are proven to be equivalent and every crisp isomorphism of two graphs is proven to be a fuzzy isomorphism. This paper provides an underlying knowledge on fuzzy graph which is an important foundation for further development and application of fuzzy graph.
References
O. Ore. Theory of Graphs. American Mathematical Society Colloquium Publications, Vol. 38, American Mathematical Society, Rhode Island, 1962, p. ix-2, retrieved from https://books.google.com.my/books?id=PfVAwAAQBAJ&q
W. K. Chen. Graph Theory and Its Engineering Applications. World Scientific Publishing, Singapore, 1997, p. 1-3.
R. J. Wilson. Introduction to Graph Theory. Longman, London, 1996, p. 3-12, retrieved from http://www.maths.ed.ac.uk/~aar/papers/wilsongraph.pdf.
M. Blue, B. Bush, J. Puckett. Applications of Fuzzy Logic to Graph Theory. Los Alamos National Laboratory: LA-UR-96-4792, 3-10, 1997.
M. Blue, B. Bush, J. Puckett. Unified Approach to Fuzzy Graph Problems. Fuzzy Sets and Systems, 125 (3), 355-668, 2002.
L. A. Zadeh. Fuzzy Sets. Information and Control, 8 (3), 338-353, 1965.
A. Rosenfeld. Fuzzy Graphs. Fuzzy Sets and Their Applications, edited by L.A. Zadeh, K.S. Fu, M. Shimura, Academic Press, New York, 77-95, 1975.
R. T. Yeh, S. Bang. Fuzzy Relations, Fuzzy Graphs, and Their Applications to Clustering Analysis. Fuzzy Sets and Their Applications, edited by L. A. Zadeh, K. S. Fu, M. Shimura, Academic Press, New York, 125-149, 1975.
T. Ahmad, S. Baharun, K. A. Arshad. Modeling a Clinical Incineration Process Using Fuzzy Autocatalytic Set. Journal of Mathematical Chemistry, 47 (4), 1263-1273, 2009.
A. N. Gani, M. B. Ahamed. Order and Size in Fuzzy Graph. Bulletin of Pure and Applied Sciences E, 22, 145-148, 2003.
G. Chartrand. Introductory Graph Theory. Dover Publications, New York, 2012, p. 32-40.
J. A. Bondy, U. S. R. Murty. Graph Theory with Applications. Vol. 290, Macmillan, London, 1976, p. 4-7, retrieved from
https://www.iro.umontreal.ca/~hahn/IFT3545/GTWA.pdf.
K. R. Bhutani. On Automorphisms of Fuzzy Graphs. Pattern Recognition Letters, 9 (3), 159-162, 1989.
A. N. Gani, J. Malarvizhi. Isomorphism on Fuzzy Graphs. International Journal of Computational and Mathematical Sciences, 2 (4), 190-196, 2008.