## Operations on fuzzy graphs.(English)Zbl 0804.05069

Different operations with graphs $$G_ 1$$ and $$G_ 2$$ yield $$G = g(G_ 1, G_ 2)$$. Under a certain hypothesis, necessary and sufficient conditions are derived for ensuring that the application of $$g$$, on two fuzzy subgraphs (fsg) of $$G_ 1$$ and $$G_ 2$$, provides a fsg of $$G$$. A proposition establishes when a fsg of $$G$$ may be represented by the application of $$g$$ on fsg’s of $$G_ i$$, $$i=1,2$$. This result is obtained when $$g$$ is the Cartesian product, the composition, the union or the join of $$G_ 1$$ and $$G_ 2$$. The authors define strong fsg’s and give conditions that ensure that such fsg’s of $$G_ i$$, $$i=1,2$$, have the same property under the use of $$g$$. Different examples are discussed.

### MSC:

 05C99 Graph theory 94C15 Applications of graph theory to circuits and networks

### Keywords:

fuzzy graph; completeness; bigraphs; operations; fuzzy subgraphs
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### References:

 [1] Bhattacharya, P., Some remarks on fuzzy graphs, Pattern Recognition Lett., 6, 297-302 (1987) · Zbl 0629.05060 [2] Bhutani, K. R., On automorphisms of fuzzy graphs, Pattern Recognition Lett., 9, 159-162 (1989) · Zbl 0800.68740 [3] Harary, F., On the group of the composition of two graphs, Duke Math. J., 26, 29-34 (1959) · Zbl 0085.37803 [4] Harary, F., Graph Theory (October 1972), Addison-Wesley: Addison-Wesley Reading, MA [5] Rosenfeld, A., Fuzzy graphs, (Zadeh, L. A.; Fu, K. S.; Shimura, M., Fuzzy Sets and Their Applications (1975), Academic Press: Academic Press New York), 77-95 [6] Sabidussi, G., The composition of graphs, Duke Math. J., 26, 693-696 (1959) · Zbl 0095.37802 [7] Sabidussi, G., Graph multiplication, Math. Z., 72, 446-457 (1960) · Zbl 0093.37603 [8] Sabidussi, G., The lexicographic product of graphs, Duke Math. J., 28, 573-578 (1961) · Zbl 0115.41102 [9] Yeh, R. T.; Bang, S. Y., Fuzzy relations, fuzzy graphs, and their applications to clustering analysis, (Zadeh, L. A.; Fu, K. S.; Shimura, M., Fuzzy Sets and Their Applications (1975), Academic Press: Academic Press New York), 125-149 · Zbl 0315.68069 [10] Zimmermann, H. J., Fuzzy Set Theory and Its Applications, (International Series in Management Science/Operations Research (1984), Kluwer-Nijhoff Publishing: Kluwer-Nijhoff Publishing Boston) · Zbl 0984.03042 [11] Zykov, A. A., Amer. Math. Soc. Transl., 79 (1952)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.