Fuzzy Graphs and Fuzzy Hypergraphs pp 83-133 | Cite as

# Applications of Fuzzy Graphs

## Abstract

Let (*V*, *μ*, *ρ*) be a fuzzy graph. We now provide two popular ways of defining the distance between a pair of vertices. One way is to define the “distance” *dis*(*x*,*y*) between *x* and *y* as the length of the shortest strongest path between them. This “distance” is symmetric and is such that *dis*(*x*,*x*) = 0 since by our definition of a fuzzy graph, no path from *x* to *x* can have strength greater than *μ*(*x*), which is the strength of the path of length 0. However, it does not satisfy the triangle property, as we see from the following example. Let *V* = {*u*, *v*, *x*, *y*,*z*}, *ρ*(*x*, *u*) = *ρ*(*u*, *v*) = *ρ*(*v*, *z*) = 1 and *ρ*(*x*, *y*) = *ρ*(*y*, *z*) = 0.5. Here any path from *x* to *y* or from *y* to *z* has strength ≤ 1/2 since it must involve either edge (*x,y*) or edge (*y*, *z*). Thus the shortest strongest paths between them have length 1. On the other hand, there is a path from *x* to *z*, through *u* and *v*, that has length 3 and strength 1. Thus *dis*(*x*,*z*) = 3 > 1 + 1 = *dis*(*x*,*y*) + *dis*(*y*, *z*) in this case.

## Keywords

Hamiltonian Path Fuzzy Subset Fuzzy Relation Connected Subgraph Fuzzy Graph## Preview

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## References

- 1.Bellman, R.E., and Zadeh, L.A.,
*Mgmt Sci*., Vol. 17, No. 4, 1970.Google Scholar - 2.Bezdek, J.C. and Harris, J.D., Fuzzy partitions and relations an axiomatic basis for clustering,
*Fuzzy Sets and Systems***1**: 111–127, 1978.MathSciNetMATHCrossRefGoogle Scholar - 3.Bhattacharya, P., Some remarks on fuzzy graphs,
*Pattern Recognition Letters***6**: 297–302, 1987.MATHCrossRefGoogle Scholar - 4.Bhattacharya, P., and Suraweera, F, An algorithm to compute the supremum of max-min powers and a property of fuzzy graphs,
*Pattern Recognition Letters***12**: 413–420, 1991.CrossRefGoogle Scholar - 5.Delgado, M. and Verdegay, J.L., On valuation and optimization problems in fuzzy graphs: A general approach and some particular cases,
*ORSA J. on Computing***2**: 74–83, 1990.MATHCrossRefGoogle Scholar - 6.Ding, B., A clustering dynamic state method for maximal trees in fuzzy graph theory,
*J. Numer. Methods Comput. Appl*.**13**: 157–160, 1992.MathSciNetGoogle Scholar - 7.Dunn, J.C., A graph theoretic analysis of pattern classification via Tamura’s fuzzy relation,
*IEEE Trans. on Systems*,*Man*,*and Cybernetics***310**–313, 1974.Google Scholar - 8.Harary, F., Graph Theoretic Methods in the Management Sciences,
*Management Science*,**5**: 387–403, 1959.MathSciNetMATHCrossRefGoogle Scholar - 9.Harary, F., and R.Z. Norman,
*Graph Theory as a Mathematical Model in Social Science*, Ann Arbor, Mich.: Institute for Social Research, 1953.Google Scholar - 10.Harary, F., R.Z. Norman and Cartwright, D.,
*Structural Models: An Introduction to the Theory of Directed Graphs*, John Wiley & Sons, Inc., New York, 1965.Google Scholar - 11.Harary, F.,
*Graph Theory*, Addison Wesley, Third printing, October 1972.Google Scholar - 12.Harary, F., and Ross, I.C., The Number of Complete Cycles in a Communication Network,
*Journal of Social Psychology*,**40**: 329–332, 1953.CrossRefGoogle Scholar - 13.Harary, F., and Ross, I.C., A Procedure for Clique Detention using the Group Matrix,
*Sociometry*,**20**: 205–215, 1957.MathSciNetCrossRefGoogle Scholar - 14.Kaufmann, A., Introduction a la Theorie des sons-ensembles flous, Vol. 1, Masson Paris, 41–189, 1973.Google Scholar
- 15.Kaufmann, A.,
*Introduction to the Theory of Fuzzy Subsets*,*Vol*.*1*, Academic Press, New York, 1975.MATHGoogle Scholar - 16.Kiss, A., An application of fuzzy graphs in database theory, Automata, languages and programming systems (Salgotarjan 1990)
*Pure Math*,*Appl. Ser. A*,**1**: 337–342, 1991.MathSciNetGoogle Scholar - 17.Kóczy, L.T., Fuzzy graphs in the evaluation and optimization of networks,
*Fuzzy Sets and Systems***46**: 307–319, 1992.MathSciNetMATHCrossRefGoogle Scholar - 18.Leenders, J.H., Some remarks on an article by Raymond T. Yeh and S.Y. Bang dealing with fuzzy relations: Fuzzy relations, fuzzy graphs, and their applications to clustering analysis, Fuzzy sets and their applications to cognitive and decision processes (Proc. U.S.-Japan Sem.,Univ. Calif., Berkeley, Calif., 1974), 125–149,
*Simon Stevin*51:93100, 1977/78.Google Scholar - 19.Ling, R.F.,
*On the theory and construction of k-cluster*, The Computer J.**15**:326–332, 1972.Google Scholar - 20.Liu, W-J., On some systems of simultaneous equations in a completely distributive lattice,
*Inform. Sci*.**50**: 185–196, 1990.MathSciNetMATHCrossRefGoogle Scholar - 21.Matula, D.W., Cluster analysis via graph theoretic techniques,
*Proc. of Lousiana Conf. on Combinatrics*,*Graph Theory*,*and Computing*, 199–212, March 1970.Google Scholar - 22.Matula, D.W.,
*k-components, clusters, and slicings in graphs*, SIAM J. Appl. Math.**22**:459–480, 1972.Google Scholar - 23.Mordeson, J.N. and Peng
*C-S, Fuzzy intersection equations*, Fuzzy Sets and Systems**60**:77–81, 1993.Google Scholar - 24.Mordeson, J.N. and Peng, C-S, Operations on fuzzy graphs,
*Inform. Sci*.**79**: 159–170, 1994.MathSciNetMATHCrossRefGoogle Scholar - 25.Mori, M. and Kawahara,
*Y., Fuzzy graph rewritings*, Theory of rewriting systems and its applications (Japanese)**918**:65–71, 1995.Google Scholar - 26.Morioka, M., Yamashita, H., and Takizawa, T., Extraction method of the difference between fuzzy graphs,
*Fuzzy information*,*knowledge representation and decision analysis*(Marseille, 1983 ), 439–444,*IFAC Proc. Ser*.,**6**, IFAC, Lexenburg, 1984.Google Scholar - 27.Nance, R.E., Korfhage, R.R., and Bhat, U.N., Information networks: Definitions and message transfer models,
*Tech. Report CP-710011*, Computer Science/Operations Research Center, SMU, Dallas, Texas, July 1971.Google Scholar - 28.Ramamoorthy, C.V., Analysis of graphs by connectivity considerations,
*JACM*,**13**: 211–222, 1966.MathSciNetMATHCrossRefGoogle Scholar - 29.Rosenblatt, D., On Linear Models and the Graphs of Minkowski–Leontief Matrices,
*Econometrica*,**25**: 325–338, 1957.MathSciNetMATHCrossRefGoogle Scholar - 30.Rosenfeld, A., Fuzzy graphs, In: L. A. Zadeh, K. S. Fu, M. Shimura, Eds.,
*Fuzzy Sets and Their Applications*,**77**–95, Academic Press, 1975.Google Scholar - 31.Ross, I.C., and Harary, F., On the Determination of Redundancies in Sociometric Chains,
*Psychometrika*,**17**: 195–208, 1952.MATHCrossRefGoogle Scholar - 32.Ross, I.C., and Harary, F., Identification of the Liaison Persons of an Organization using the Structure Matrix,
*Management Science*,**1**: 251–258, 1955.MathSciNetMATHCrossRefGoogle Scholar - 33.Ross, I.C., and Harary, F., A Description of Strengthening and Weakening Members of a Group,
*Sociometry*,**22**: 139–147, 1959.MathSciNetCrossRefGoogle Scholar - 34.Sibson, R., Some observation on a paper by Lance and Williams,
*The Computer J*.**14**: 156–157, 1971.MATHCrossRefGoogle Scholar - 35.Sunouchi, H. and Morioka, M., Some properties on the connectivity of a fuzzy graph (Japanese),
*Bull. Sci. Engrg. Res. lab. Waseda Univ*. no. 132, 70–78, 1991.MathSciNetGoogle Scholar - 36.Takeda, E., Connectvity in fuzzy graphs,
*Tech. Rep. Osaka Univ*.**23**: 343–352, 1973.MathSciNetGoogle Scholar - 37.Takeda, E. and Nishida, T., An application of fuzzy graph to the problem concerning group structure,
*J. Operations Res. Soc. Japan***19**: 217–227, 1976.MathSciNetMATHGoogle Scholar - 38.Tong, Z. and Zheng, D., An algorithm for finding the connectedness matrix of a fuzzy graph,
*Congr. Numer*.**120**: 189–192, 1996.MathSciNetMATHGoogle Scholar - 39.Ullman, J. D.,
*Principles of Database and Knowledge-base Systems*, Vol 1–2, Computer Science Press, Rockville, MD., 1989.Google Scholar - 40.Wu, L. G. and Chen, T.P., Some problems concerning fuzzy graphs (Chinese),
*J. Huazhong Inst. Tech*. no 2, Special issue on fuzzy math, iv, 58–60, 1980.Google Scholar - 41.Xu, J., The use of fuzzy graphs in chemical structure research, In: D.H. Rouvry, Ed.,
*Fuzzy Logic in Chemistry*, 249–282, Academic Press, 1997.Google Scholar - 42.Yamashita, H., Approximation algorithm for a fuzzy graph (Japanese),
*Bull. Centre Info*, ru.**2**: 59–60, 1985.Google Scholar - 43.Yamashita, H., Structure analysis of fuzzy graph and its application (Japanese),
*Bull. Sci. Engrg. Res. Lab. Waseda Univ*. no.**132**, 61–69, 1991.Google Scholar - 44.Yamashita, H. and Morioka, M., On the global structure of a fuzzy graph,
*Analysis of Fuzzy Information*,**1**:167–176, CRC, Boca Raton, Fla., 1987.Google Scholar - 45.Yeh, R.T. and Bang, S.Y., Fuzzy relations, fuzzy graphs, and their applications to clustering analysis, In: L. A. Zadeh, K. S. Fu, M. Shimura, Eds.,
*Fuzzy Sets and Their Applications*, 125–149, Academic Press, 1975.Google Scholar - 46.Zadeh, L.A., Fuzzy Sets,
*Information and Control*,**8**: 338–353, 1965.MathSciNetMATHCrossRefGoogle Scholar - 47.Zadeh, L.A., Similarity relations and fuzzy orderings,
*Information Sciences*,**3**: 177–200, 1971.MathSciNetMATHCrossRefGoogle Scholar - 48.Zhu, R.Y., The critical number of the connectivity degree of a fuzzy graph (Chineses),
*Fuzzy Math*.**2**: 113–116, 1982.Google Scholar - 49.Zykov, A.A., On some properties of linear complexes (Russian),
*Mat. Sbornik***24**:163–188, 1949,*Amer. Math. Soc. Translations N*. 79, 1952.Google Scholar