On M-strong fuzzy graphs

The cartesian product and disjoint sum of graphs play a prominent role and have numerous interesting algebraic properties. In this note, we consider operations on fuzzy graphs under which M-strong property is preserved. If G1 and G2 are M-strong fuzzy graphs then we prove that G1 × G2, G1 [G2] and G1 + G2 are also M-strong but G1 ∪ G2 need not be M-strong. If G1 × G2 is M-strong then we show that at least one factor must be M-strong. We show that the product of a M-strong fuzzy graph G1 with a non-M-strong fuzzy graph G2 remains M-strong if and only if G2 satisfies special condition. For any fuzzy graph G, Gcc is the smallest M-strong fuzzy graph that contains G and G = Gcc if and only if G is M-strong. We further show that M-strong fuzzy graph G is a fuzzy tree if and only if the support(G) is a tree.