2D Structured Non-Cyclic Fuzzy Graphs

Authors: T. Pathinathan, M. Peter

Abstract:

Fuzzy graphs incorporate concepts from graph theory with fuzzy principles. In this paper, we make a study on the properties of fuzzy graphs which are non-cyclic and are of two-dimensional in structure. In particular, this paper presents 2D structure or the structure of double layer for a non-cyclic fuzzy graph whose underlying crisp graph is non-cyclic. In any graph structure, introducing 2D structure may lead to an inherent cycle. We propose relevant conditions for 2D structured non-cyclic fuzzy graphs. These conditions are extended even to fuzzy graphs of the 3D structure. General theoretical properties that are studied for any fuzzy graph are verified to 2D structured or double layered fuzzy graphs. Concepts like Order, Degree, Strong and Size for a fuzzy graph are studied for 2D structured or double layered non-cyclic fuzzy graphs. Using different types of fuzzy graphs, the proposed concepts relating to 2D structured fuzzy graphs are verified.

Keywords: Double layered fuzzy graph, double layered non-cyclic fuzzy graph, strong, order, degree and size.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1317210

References:

[1] L. A. Zadeh, “Fuzzy sets”, Information and Control, 1965, pp. 338-353.
[2] A. Rosenfeld, “Fuzzy Graphs”, in L. A. Zadeh, K. S. Fu, K. Tanaka and M. Shimura, eds., Fuzzy Sets and Their Applications to Cognitive and Decision Process, Academic Press, New York, 1975, pp. 75-95.
[3] R. T. Yeh and S. Y. Bang, “Fuzzy Relations, Fuzzy Graphs and their Applications to Clustering Analysis”, in L. A. Zadeh, K. S. Fu, K. Tanaka and M. Shimura, eds., Fuzzy Sets and Their Applications to Cognitive and Decision Process, Academic Press, New York, 1975, pp. 75-95.
[4] M. S. Sunita and A. Vijaya Kumar, “Complement of Fuzzy Graph”, Indian Journal of Pure and Applied Mathematics, vol. 33, no. 9, 2002, pp. 1451-1464.
[5] A. Nagoorgani and J. Malarvizhi, “Properties of - Complement of a Fuzzy Graph”, International Journal of Algorithms, Computing and Mathematics, vol. 2, no. 3, 2009, pp. 73-83.
[6] A. Nagoorgani and K. Radha, “The Degree of Vertex in some Fuzzy Graphs”, International Journal of Algorithms, Computing and Mathematics, vol. 2, no. 3, 2009, pp. 107-116.
[7] T. Pathinathan and J. Jesintha Roseline, “Double Layered Fuzzy Graph”, Annals of Pure and Applied Mathematics, vol. 8, no. 1, 2014, pp. 135-143.
[8] T. Pathinathan and J. Jesintha Roseline, “Matrix Representation of Double Layered Fuzzy Graph and its Properties”, Annals of Pure and Applied Mathematics, vol. 8, no. 2, 2014, pp. 51-58.
[9] T. Pathinathan and J. Jesintha Roseline, “Vertex Degree of Cartesian product of Intuitionistic Fuzzy Graph”, International Journal of Scientific and Engineering Research, vol. 5, no. 9, 2014, pp. 224-227.
[10] K. T. Atanassov, Intuitionistic Fuzzy Sets, Fuzzy Sets and Systems, vol. 20, 1986, pp.87-96.
[11] T. Pathinathan and J. Jesintha Roseline, “Intuitionistic double Layered Fuzzy Graph and its Properties”, Asian Research Publishinng Network, 2015, pp. 5413-5417.
[12] T. Pathinathan and J. Jesintha Roseline, “Intuitionistic double Layered Fuzzy Graph and its Cartesian product Vertex degree”, International Journal of Computing Algorithm, 2015, pp. 1374-1378.
[13] T. Pathinathan and J. Jesintha Roseline, “Structural Core Graph of double Layered Fuzzy Graph”, International Journal of Fuzzy Mathematical Archives, vol. 8, no. 2, 2015, pp. 59-67.
[14] T. Pathinathan and J. Jesintha Roseline, “Triple Layered Fuzzy graph”, International Journal of Fuzzy Mathematical Archives, vol. 8, no.1, 2015, pp. 36-42.
[15] T. Pathinathan and M. Peter, “Balanced Double Layered Fuzzy Graph”, International Journal of Multidisciplinary Research and Modern Education (IJMRME), vol. 3, no. 1, 2017, pp. 208-217.
[16] M. Peter, “On Co-Normal Product of two Fuzzy Graphs”, International Journal of Multidisciplinary Research and Modern Education (IJMRME), vol. 3, no. 1, 2017, pp. 203-207.