Fuzzy Graphs and Fuzzy Hypergraphs

by Mordeson, John N.; Nair, Premchand S.

• ISBN13: 9783790812862
• ISBN10: 3790812862
• Format: Hardcover
• Copyright: 2000-04-01
• Publisher: Physica Verlag

Summary

The authors present an up-to-date account of results from fuzzy graph theory and fuzzy hypergraph theory and give applications of the results. The book should be of interest to research mathematicians and to engineers and computer scientists interested in applications. Some specific application areas presented from fuzzy graph theory are cluster analysis, pattern classfication, database theory, and the problem concerning group structure. Applications of fuzzy hypergraph theory to portfolio management, managerial decision making with an example to waste management, and to neural cell-assemblies are given. It is shown how (fuzzy) hypergraphs and rough sets are related in such a way that ideas may be carried back and forth between the two areas.

Table of Contents

Foreword	v
Preface	vii
Acknowledgments	ix
Fuzzy Subsets	1	(18)
Fuzzy Relations	4	(2)
Fuzzy Equivalence Relations	6	(3)
Pattern Classification	9	(2)
Similarity Relations	11	(5)
References	16	(3)
Fuzzy Graphs	19	(64)
Paths and Connectedness	20	(20)
Bridges and Cut Vertices	21	(1)
Forests and Trees	22	(3)
Trees and Cycles	25	(1)
A Characterization of Fuzzy Trees	26	(5)
(Fuzzy) Cut Sets	31	(2)
(Fuzzy) Chords, (Fuzzy) Cotrees, and (Fuzzy) Twigs	33	(2)
(Fuzzy) 1-Chain with Boundary 0, Coboundary, and Cocyles	35	(1)
(Fuzzy) Cycle Set and (Fuzzy) Cocycle Set	36	(4)
Fuzzy Line Graphs	40	(5)
Fuzzy Interval Graphs	45	(17)
Fuzzy Intersection Graphs	46	(1)
Fuzzy Interval Graphs	47	(2)
The Fulkerson and Gross Characterization	49	(2)
The Gilmore and Hoffman Characterization	51	(11)
Operations on Fuzzy Graphs	62	(8)
Cartesian Product and Composition	62	(4)
Union and Join	66	(4)
On Fuzzy Tree Definition	70	(8)
References	78	(5)
Applications of Fuzzy Graphs	83	(52)
Clusters	86	(1)
Cluster Analysis	87	(10)
Cohesiveness	92	(1)
Slicing in Fuzzy Graphs	93	(4)
Application to Cluster Analysis	97	(5)
Fuzzy Intersection Equations	102	(6)
Existence of Solutions	102	(6)
Fuzzy Graphs in Database Theory	108	(4)
Representation of Dependency Structure λ(X, Y)	111	(1)
A Description of Strengthening and Weakening Members of a Group	112	(8)
Connectedness Criteria	114	(3)
Inclusive Connectedness Categories	117	(1)
Exclusive Connectedness Categories	118	(2)
An Application to the Problem Concerning Group Structure	120	(9)
Connectedness of a Fuzzy Graph	121	(2)
Weakening and Strengthening Points a Directed Graph	123	(6)
References	129	(6)
Fuzzy Hypergraphs	135	(98)
Fuzzy Hypergraphs	135	(6)
Fuzzy Transversals of Fuzzy Hypergraphs	141	(29)
Properties of Tr (H)	153	(5)
Construction of Hs	158	(12)
Coloring of Fuzzy Hypergraphs	170	(29)
β-degree Coloring Procedures	179	(15)
Chromatic Values of Fuzzy Colorings	194	(5)
Intersecting Fuzzy Hypergraphs	199	(18)
Characterization of Strongly Intersecting Hypergraphs	205	(2)
Simply Ordered Intersecting Hypergraphs	207	(5)
H-dominant Transversals	212	(5)
Hebbian Structures	217	(4)
Additional Applications	221	(8)
References	229	(4)
List of Figures	233	(2)
List of Tables	235	(2)
List of Symbols	237	(6)
Index	243