Advertisement

Applications of Fuzzy Graphs

  • John N. Mordeson
  • Premchand S. Nair
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 46)

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bellman, R.E., and Zadeh, L.A., Mgmt Sci., Vol. 17, No. 4, 1970.Google Scholar
  2. 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. 3.
    Bhattacharya, P., Some remarks on fuzzy graphs, Pattern Recognition Letters 6: 297–302, 1987.MATHCrossRefGoogle Scholar
  4. 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. 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. 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. 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. 8.
    Harary, F., Graph Theoretic Methods in the Management Sciences, Management Science, 5: 387–403, 1959.MathSciNetMATHCrossRefGoogle Scholar
  9. 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. 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. 11.
    Harary, F., Graph Theory, Addison Wesley, Third printing, October 1972.Google Scholar
  12. 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. 13.
    Harary, F., and Ross, I.C., A Procedure for Clique Detention using the Group Matrix, Sociometry, 20: 205–215, 1957.MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kaufmann, A., Introduction a la Theorie des sons-ensembles flous, Vol. 1, Masson Paris, 41–189, 1973.Google Scholar
  15. 15.
    Kaufmann, A., Introduction to the Theory of Fuzzy Subsets, Vol. 1, Academic Press, New York, 1975.MATHGoogle Scholar
  16. 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. 17.
    Kóczy, L.T., Fuzzy graphs in the evaluation and optimization of networks, Fuzzy Sets and Systems 46: 307–319, 1992.MathSciNetMATHCrossRefGoogle Scholar
  18. 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. 19.
    Ling, R.F., On the theory and construction of k-cluster, The Computer J. 15:326–332, 1972.Google Scholar
  20. 20.
    Liu, W-J., On some systems of simultaneous equations in a completely distributive lattice, Inform. Sci. 50: 185–196, 1990.MathSciNetMATHCrossRefGoogle Scholar
  21. 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. 22.
    Matula, D.W., k-components, clusters, and slicings in graphs, SIAM J. Appl. Math. 22:459–480, 1972.Google Scholar
  23. 23.
    Mordeson, J.N. and PengC-S, Fuzzy intersection equations, Fuzzy Sets and Systems 60:77–81, 1993.Google Scholar
  24. 24.
    Mordeson, J.N. and Peng, C-S, Operations on fuzzy graphs, Inform. Sci. 79: 159–170, 1994.MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Mori, M. and Kawahara, Y., Fuzzy graph rewritings, Theory of rewriting systems and its applications (Japanese) 918:65–71, 1995.Google Scholar
  26. 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. 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. 28.
    Ramamoorthy, C.V., Analysis of graphs by connectivity considerations, JACM, 13: 211–222, 1966.MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Rosenblatt, D., On Linear Models and the Graphs of Minkowski–Leontief Matrices, Econometrica, 25: 325–338, 1957.MathSciNetMATHCrossRefGoogle Scholar
  30. 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. 31.
    Ross, I.C., and Harary, F., On the Determination of Redundancies in Sociometric Chains, Psychometrika, 17: 195–208, 1952.MATHCrossRefGoogle Scholar
  32. 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. 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. 34.
    Sibson, R., Some observation on a paper by Lance and Williams, The Computer J. 14: 156–157, 1971.MATHCrossRefGoogle Scholar
  35. 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. 36.
    Takeda, E., Connectvity in fuzzy graphs, Tech. Rep. Osaka Univ. 23: 343–352, 1973.MathSciNetGoogle Scholar
  37. 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. 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. 39.
    Ullman, J. D., Principles of Database and Knowledge-base Systems, Vol 1–2, Computer Science Press, Rockville, MD., 1989.Google Scholar
  40. 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. 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. 42.
    Yamashita, H., Approximation algorithm for a fuzzy graph (Japanese), Bull. Centre Info, ru. 2: 59–60, 1985.Google Scholar
  43. 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. 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. 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. 46.
    Zadeh, L.A., Fuzzy Sets, Information and Control, 8: 338–353, 1965.MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    Zadeh, L.A., Similarity relations and fuzzy orderings, Information Sciences, 3: 177–200, 1971.MathSciNetMATHCrossRefGoogle Scholar
  48. 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. 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

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • John N. Mordeson
    • 1
  • Premchand S. Nair
    • 2
  1. 1.Center for Research in Fuzzy Mathematics and Computer ScienceOmahaUSA
  2. 2.Department of Mathematics and Computer ScienceCreighton UniversityOmahaUSA

Personalised recommendations