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**Node connectivity and arc connectivity of a fuzzy graph.**
*(English)*
Zbl 1233.05163

The fuzzy graph approach is more powerful in cluster analysis than the usual graph-theoretic approach due to its ability to handle the strengths of arcs effectively. It is shown that the minimum and maximum degree can be represented by only node strength in a complete fuzzy graph in Section 3 of this paper. In Section 4, the authors introduce the concept of node-strength sequence and study four classes of node-strength sequences of complete fuzzy graphs with respect to minimum and maximum strong degree. Two new connectivity parameters in fuzzy graphs, namely fuzzy node connectivity (\(\kappa (G)\)) and fuzzy arc connectivity (\(\kappa'(G)\)) of a connected fuzzy graph \(G\) are introduced in Sections 5 and Section 6, respectively. Fuzzy node cut, fuzzy arc cut and fuzzy bond are defined in Sections 5 and 6, too. Fuzzy bond is a special type of a fuzzy bridge. It is proved that at least one of the end nodes of a fuzzy bond is a fuzzy cut-node. In Section 7, the authors obtain the fuzzy analogue of Whitney’s theorem as follows: \(\kappa (G)\leq \kappa {^{\prime }}(G)\leq \delta_S(G)\), where \(\delta_S(G)\) is the minimum strong degree of fuzzy graph \(G\). It is shown that \(\kappa (G)=\kappa'(G)\) for a fuzzy tree and it is the minimum of the strengths of its strong arcs. The relationships of the new parameters with already existing vertex and edge connectivity parameters are studied and it is shown that the values of all these parameters are equal in a complete fuzzy graph in Section 8. Also a new clustering technique based on fuzzy arc connectivity is introduced in Section 9.

Reviewer: Kewen Zhao (Hainan)

### MSC:

05C72 | Fractional graph theory, fuzzy graph theory |

05C40 | Connectivity |

03E72 | Theory of fuzzy sets, etc. |

05C22 | Signed and weighted graphs |

### Keywords:

fuzzy relation; fuzzy bonds; fuzzy node cut; fuzzy arc cut; fuzzy node connectivity; fuzzy arc connectivity
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\textit{S. Mathew} and \textit{M. S. Sunitha}, Inf. Sci. 180, No. 4, 519--531 (2010; Zbl 1233.05163)

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