Book ChapterDOI
Fuzzy relations, fuzzy graphs, and their applications to clustering analysis
Raymond T. Yeh,S.Y. Bang +1 more
- pp 125-149
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TLDR
In this article, the concept of similarity relation introduced by L. A. Zadeh is derivable in much the same way as equivalence relation, and the resolution identity is brought out quite naturally.Abstract:
Publisher Summary This chapter discusses the basic terminologies and notations regarding fuzzy relations and fuzzy graphs. The the concept of similarity relation introduced by L. A. Zadeh is derivable in much the same way as equivalence relation. Moreover, through this derivation, the resolution identity is brought out quite naturally. The chapter analyzes fuzzy graphs from the connectedness viewpoint and the presents the application of results to clustering analysis and modeling of information networks. The usual graph-theoretical approaches to clustering analysis involve first obtaining a threshold graph from a fuzzy graph and then applying various techniques to obtain clusters as maximal components under different connectivity considerations. These methods have a common weakness, namely, the weight of edges are not treated fairly because any weight greater (less) than the threshold is treated as 1(0).read more
Citations
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Journal ArticleDOI
On the structure of F-indistinguishability operators
TL;DR: It is shown that any of such operators is generated by a family of fuzzy subsets, which gives the way to construct F-indistinguishabilities, and facilitates new applications of fuzzy relations.
Journal ArticleDOI
Some remarks on fuzzy graphs
TL;DR: It is shown that a fuzzy group can be associated with a fuzzy graph in a natural way and some properties of fuzzy graphs are considered and the notions of eccentricity and center are introduced.
Journal ArticleDOI
Fuzzy partitions and relations; an axiomatic basis for clustering
James C. Bezdek,J Douglas Harris +1 more
TL;DR: A new definition of transitivity for fuzzy relations yields a relation-theoretic characterization of the class of all psuedo-metrics on a fixed data set into the closed unit interval.
Journal ArticleDOI
Operations on fuzzy graphs
John N. Mordeson,Chang-Shyh Peng +1 more
TL;DR: If the graph G is formed from G1 and G2 by one of the operations of Cartesian product, composition, union, and join, then necessary and sufficient conditions for an arbitrary fuzzy subgraph of G also to be formed by the same operation from fuzzy sub graphs of G2.
Journal ArticleDOI
m-Polar fuzzy sets: an extension of bipolar fuzzy sets.
TL;DR: It is proved that bipolar fuzzy sets and [0,1]2-sets (which have been deeply studied) are actually cryptomorphic mathematical notions and many results which are related to these concepts can be generalized to the case of m-polar fuzzy sets.
References
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Journal ArticleDOI
Similarity relations and fuzzy orderings
TL;DR: An extended version of Szpilrajn's theorem is proved and various properties of similarity relations and fuzzy orderings are investigated and, as an illustration, a fuzzy preordering is investigated which is reflexive and antisymmetric.
Journal ArticleDOI
A General Theory of Classificatory Sorting Strategies 1. Hierarchical Systems
G. N. Lance,W. T. Williams +1 more
TL;DR: It is shown that the computational behaviour of a hierarchical sorting-strategy depends on three properties, which are established for five conventional strategies and four measures.
Journal ArticleDOI
Abstraction and pattern classification
TL;DR: In this article, a general framework for the treatment of pattern recognition problems is discussed, in which the notion of a "fuzzy" set is introduced and used to determine whether a symbol is a member of a particular set or not.
Journal ArticleDOI
Pattern Classification Based on Fuzzy Relations
TL;DR: A method of classifying patterns using fuzzy relations is described, where a similitude between any two patterns is calculated by using the composition of a fuzzy relation to induce an equivalence relation.