Journal of Applied Mathematics

Certain Types of Interval-Valued Fuzzy Graphs

Muhammad Akram, Noura Omair Alshehri, and Wieslaw A. Dudek

Full-text: Open access

Abstract

We propose certain types of interval-valued fuzzy graphs including balanced interval-valued fuzzy graphs, neighbourly irregular interval-valued fuzzy graphs, neighbourly total irregular interval-valued fuzzy graphs, highly irregular interval-valued fuzzy graphs, and highly total irregular interval-valued fuzzy graphs. Some interesting properties associated with these new interval-valued fuzzy graphs are investigated, and necessary and sufficient conditions under which neighbourly irregular and highly irregular interval-valued fuzzy graphs are equivalent are obtained. We also describe the relationship between intuitionistic fuzzy graphs and interval-valued fuzzy graphs.

Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 857070, 11 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394808175

Digital Object Identifier
doi:10.1155/2013/857070

Mathematical Reviews number (MathSciNet)
MR3122137

Zentralblatt MATH identifier
06950910

Citation

Akram, Muhammad; Alshehri, Noura Omair; Dudek, Wieslaw A. Certain Types of Interval-Valued Fuzzy Graphs. J. Appl. Math. 2013 (2013), Article ID 857070, 11 pages. doi:10.1155/2013/857070. https://projecteuclid.org/euclid.jam/1394808175


Export citation

References

  • S. G. Shirinivas, S. Vetrivel, and N. M. Elango, “Applications of graph theory in computer science–-an overview,” International Journal of Engineering Science and Technology, vol. 2, no. 9, pp. 4610–4621, 2010.
  • N. Deo, Graph Theory with Applications to Engineering and Computer Science, Prentice Hall, Englewood Cliffs, NJ, USA, 1990.
  • L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning–-I,” Information Sciences, vol. 8, pp. 199–249, 1975.
  • K. T. Atanassov, “Intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 20, no. 1, pp. 87–96, 1986.
  • L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, pp. 338–353, 1965.
  • J. M. Mendel, Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions, Prentice-Hall, Upper Saddle River, NJ, USA, 2001.
  • M. B. Gorzałczany, “A method of inference in approximate reasoning based on interval-valued fuzzy sets,” Fuzzy Sets and Systems, vol. 21, no. 1, pp. 1–17, 1987.
  • M. B. Gorzałczany, “An interval-valued fuzzy inference method–-some basic properties,” Fuzzy Sets and Systems, vol. 31, no. 2, pp. 243–251, 1989.
  • M. K. Roy and R. Biswas, “I-V fuzzy relations and Sanchez's approach for medical diagnosis,” Fuzzy Sets and Systems, vol. 47, no. 1, pp. 35–38, 1992.
  • I. B. Türksen, “Interval valued fuzzy sets based on normal forms,” Fuzzy Sets and Systems, vol. 20, no. 2, pp. 191–210, 1986.
  • A. Kauffman, Introduction a la Theorie des Sous-emsembles Flous, vol. 1, Masson et Cie, 1973.
  • A. Rosenfeld, “Fuzzy graphs,” in Fuzzy Sets and Their Applications, L. A. Zadeh, K. S. Fu, and M. Shimura, Eds., pp. 77–95, Academic Press, New York, NY, USA, 1975.
  • J. N. Mordeson and C.-S. Peng, “Operations on fuzzy graphs,” Information Sciences, vol. 79, no. 3-4, pp. 159–170, 1994.
  • M. S. Sunitha and A. V. Kumar, “Complement of a fuzzy graph,” Indian Journal of Pure and Applied Mathematics, vol. 33, no. 9, pp. 1451–1464, 2002.
  • H. Ju and L. Wang, “Interval-valued fuzzy subsemigroups and subgroups associated by intervalvalued fuzzy graphs,” in Proceedings of the WRI Global Congress on Intelligent Systems (GCIS '09), pp. 484–487, Xiamen, China, May 2009.
  • M. Akram, “Bipolar fuzzy graphs,” Information Sciences, vol. 181, no. 24, pp. 5548–5564, 2011.
  • M. Akram, “Bipolar fuzzy graphs with applications,” Knowledge Based Systems, vol. 39, pp. 1–8, 2013.
  • M. Akram, “Interval-valued fuzzy line graphs,” Neural Computing and Applications, vol. 21, pp. 145–150, 2012.
  • M. Akram and W. A. Dudek, “Interval-valued fuzzy graphs,” Computers & Mathematics with Applications, vol. 61, no. 2, pp. 289–299, 2011.
  • M. Akram and B. Davvaz, “Strong intuitionistic fuzzy graphs,” Filomat, vol. 26, no. 1, pp. 177–196, 2012.
  • M. Akram, K. H. Dar, and K. P. Shum, “Interval-valued ($\alpha $,$\beta $)-fuzzy \emphK-algebras,” Applied Soft Computing Journal, vol. 11, no. 1, pp. 1213–1222, 2011.
  • T. AL-Hawary, “Complete fuzzy graphs,” International Journal of Mathematical Combinatorics, vol. 4, pp. 26–34, 2011.
  • P. Bhattacharya, “Some remarks on fuzzy graphs,” Pattern Recognition Letters, vol. 6, no. 5, pp. 297–302, 1987.
  • S. M. Chen, “Interval-valued fuzzy hypergraph and fuzzy partition,” IEEE Transactions on Systems, Man, and Cybernetics B, vol. 27, no. 4, pp. 725–733, 1997.
  • G. Deschrijver and C. Cornelis, “Representability in interval-valued fuzzy set theory,” International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, vol. 15, no. 3, pp. 345–361, 2007.
  • A. N. Gani and S. R. Latha, “On irregular fuzzy graphs,” Applied Mathematical Sciences, vol. 6, no. 9–12, pp. 517–523, 2012.
  • X. Ma, J. Zhan, B. Davvaz, and Y. B. Jun, “Some kinds of $(\in ,\in \vee q)$-interval-valued fuzzy ideals of BCI-algebras,” Information Sciences, vol. 178, no. 19, pp. 3738–3754, 2008.
  • J. N. Mordeson and P. S. Nair, Fuzzy Graphs and Fuzzy Hypergraphs, Studies in Fuzziness and Soft Computing, Physica, Heidelberg, Germany, 1998.
  • F. Riaz and K. M. Ali, “Applications of graph theory in computer science,” in Proceedings of the 3rd International Conference on Computational Intelligence, Communication Systems and Networks (CICSyN '11), pp. 142–145, Bali, Indonesia, July 2011.
  • M. Behzad and G. Chartrand, “No graph is perfect,” The American Mathematical Monthly, vol. 74, pp. 962–963, 1967.
  • G. Deschrijver and E. E. Kerre, “On the relationship between some extensions of fuzzy set theory,” Fuzzy Sets and Systems, vol. 133, no. 2, pp. 227–235, 2003.
  • A. Shannon and K. T. Atanassov, “A first step to a theory of the intuitionistic fuzzy graphs,” in Proceeding of the FUBEST, D. Lakov, Ed., pp. 59–61, Sofia, Bulgaria, 1994.
  • R. Parvathi, M. G. Karunambigai, and K. T. Atanassov, “Operations on intuitionistic fuzzy graphs,” in Proceedings of the IEEE International Conference on Fuzzy Systems, pp. 1396–1401, August 2009.