Abstract
The aim of the present paper is to show usefulness of a class of fuzzy graphs as a basis of hierarchical clustering. For this purpose classical results are reviewed and new results are proved using fuzzy graphs. Namely, the followings are discussed. 1. A formalization of hierarchical classifications is given and its relation with traditional description of methods in hierarchical clustering is discussed. 2. Meaning of the fuzzy graphs employed herein is explained in terms of a three dimensional description. 3. Equivalence between the Wishart method of mode analysis and connected components of fuzzy graphs with fuzzy grades on the vertices is proved. 4. Refinement relation between two methods of hierarchical clustering is defined and it is proved that this relation holds for the Ling method and the Wishart method and for the nearest neighbor method and the group average method(or the farthest neighbor method). 5. Significance of the results of equivalence and refinement in applications are discussed. The above 3,4,and 5 show new results, whereas 1 and 2 are reviews. To compile old results into the present form is necessary in order to describe new results in an appropriate way. In this sense, the review is an essential part of the present paper.
