## Fuzzy Graphs and Fuzzy HypergraphsIn the course of fuzzy technological development, fuzzy graph theory was identified quite early on for its importance in making things work. Two very important and useful concepts are those of granularity and of nonlinear ap proximations. The concept of granularity has evolved as a cornerstone of Lotfi A.Zadeh's theory of perception, while the concept of nonlinear approx imation is the driving force behind the success of the consumer electronics products manufacturing. It is fair to say fuzzy graph theory paved the way for engineers to build many rule-based expert systems. In the open literature, there are many papers written on the subject of fuzzy graph theory. However, there are relatively books available on the very same topic. Professors' Mordeson and Nair have made a real contribution in putting together a very com prehensive book on fuzzy graphs and fuzzy hypergraphs. In particular, the discussion on hypergraphs certainly is an innovative idea. For an experienced engineer who has spent a great deal of time in the lab oratory, it is usually a good idea to revisit the theory. Professors Mordeson and Nair have created such a volume which enables engineers and design ers to benefit from referencing in one place. In addition, this volume is a testament to the numerous contributions Professor John N. Mordeson and his associates have made to the mathematical studies in so many different topics of fuzzy mathematics. |

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### Contents

IV | 1 |

V | 4 |

VI | 6 |

VII | 9 |

VIII | 11 |

IX | 16 |

X | 19 |

XI | 20 |

XXXVI | 97 |

XXXVII | 102 |

XXXVIII | 108 |

XXXIX | 111 |

XL | 112 |

XLI | 114 |

XLII | 117 |

XLIII | 118 |

XII | 21 |

XIII | 22 |

XIV | 25 |

XV | 26 |

XVI | 31 |

XVII | 33 |

XVIII | 35 |

XIX | 36 |

XX | 40 |

XXI | 45 |

XXII | 46 |

XXIII | 47 |

XXIV | 49 |

XXV | 51 |

XXVI | 62 |

XXVIII | 66 |

XXIX | 70 |

XXX | 78 |

XXXI | 83 |

XXXII | 86 |

XXXIII | 87 |

XXXIV | 92 |

XXXV | 93 |

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### Common terms and phrases

acyclic assume C-coloring Cartesian product cliques of G cocycle coloring of H complete fuzzy connected connectedness construction core hypergraph Corollary crisp hypergraph cut level Define the fuzzy Definition denote digraph directed graph edge of H edge set element equivalence relation Example exists function fuzzy cliques fuzzy coloring fuzzy cycle fuzzy edge fuzzy forest fuzzy graph fuzzy interval graph fuzzy relation Fuzzy Sets fuzzy subset fuzzy tree graph G graph theory Hence implies incidence matrix intersection graph Lemma Let G Let H line graph linear ordering maximal MH(X minimal fuzzy transversal minimal transversal nontrivial ordered fuzzy hypergraph pair partial fuzzy subgraph partition phase sequence Proposition reflexive result rough set satisfies sequentially simple simply ordered strength strongest path strongly intersecting subgraph of G supp(p symmetric t-cut T-related Theorem Tr(H Tr(Ti transitive orientation transversal of H vertex set vertices