Weighted Graphs and Fuzzy Graphs
محورهای موضوعی : Transactions on Fuzzy Sets and Systems
John Mordeson
1
,
Anntreesa Josy
2
,
Sunil Mathew
3
1 - Department of Mathematics, Creighton University, Nebraska, USA.
2 - Department of Mathematics, National Institute of Technology Calicut, Calicut, India.
3 - Department of Mathematics, National Institute of Technology Calicut, Calicut, India.
کلید واژه: Weighted graphs, Fuzzy graphs, Isomorphic lattices,
چکیده مقاله :
It has been shown in [3] that in the two-dimensional case, the lattices of truth values considered are pairwise isomorphic, and so are the corresponding families of fuzzy sets. Therefore, each result for one of these types of fuzzy sets can be directly rewritten for each (isomorphic) type of fuzzy sets. In this paper, we show that there is a strong connection between weighted graphs and fuzzy graphs. We accomplish this by using lattice isomorphisms. Consequently, under certain conditions, results for one area can be carried over immediately to the other. Many situations in fuzzy graph theory do not depend on the weights of the vertices. The situation of providing weights for the vertices of a weighted graph is also considered. We also consider lattice homomorphisms with an illustration involving nonstandard analysis. In particular, we consider a nonstandard weighted graph, i.e., a graph where the weights of the edges are from a nonstandard interval.
It has been shown in [3] that in the two-dimensional case, the lattices of truth values considered are pairwise isomorphic, and so are the corresponding families of fuzzy sets. Therefore, each result for one of these types of fuzzy sets can be directly rewritten for each (isomorphic) type of fuzzy sets. In this paper, we show that there is a strong connection between weighted graphs and fuzzy graphs. We accomplish this by using lattice isomorphisms. Consequently, under certain conditions, results for one area can be carried over immediately to the other. Many situations in fuzzy graph theory do not depend on the weights of the vertices. The situation of providing weights for the vertices of a weighted graph is also considered. We also consider lattice homomorphisms with an illustration involving nonstandard analysis. In particular, we consider a nonstandard weighted graph, i.e., a graph where the weights of the edges are from a nonstandard interval.
[1] I. Goldbring, Lecture Notes on Nonstandard Analysis, UCLA Summer School in Logic, (2014).
[2] T. Imamura, Note on the de nition of neutrosophic logic, Journal of Japan Society for Fuzzy Theory and Intelligent Informatics, 34(3) (2022), 669-672.
[3] E. P. Klement and R. Mesiar, L-fuzzy sets and isomorphic lattices: are all the \new" results really new? Mathematics, 6(9) (2018), 1-24.
[4] G. J. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Applications, Prentice Hall PTR, New Jersey, (1995).
[5] J. N. Mordeson, S. Mathew, and G. Gayathri, Fuzzy Graph Theory: Applications to Global Problems, Springer, to appear.
[6] G. A. Robinson, Nonstandard Analysis, North-Holland Publishing Co., (1966).
[7] A. Rosenfeld, Fuzzy graphs, In L. A. Zadeh, K. S. Fu, and M. Shimura, Eds., Fuzzy Sets and Their Applications, Academic Press, New York, (1975), 77-95.
[8] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.
