Operations on fuzzy graphs (Q1331033)

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.