Arc disjoint fuzzy graphs

Examines a special kind of fuzzy graph called an arc-disjoint graph. Let S be a finite set, σ a function from S into the closed interval [0,1], and μ a function of S×S into [0,1] such that μ(x,y)=μ(y,x) for all x,y in S. Then the pair G=(σ,μ) is called a fuzzy graph if μ(x,y)⩽min{σ(x),σ(y)}. Let G=(σ,μ) be a fuzzy graph. Define the function μ^∞ of S×S into [0,1] by ∀x,y∈S, μ^∞ (x,y)=sup{μⁱ(x,y) | i=1, 2, ...}, where μⁱ is the max-min composition of μ with itself i times (i=1, 2, ...). G is called arc-disjoint if no two cycles share a common arc. We show that if G is arc-disjoint, then G is a fuzzy forest iff in any cycle C of G, there is an arc (x,y) such that μ(x,y)<μ(u,v) where (u,v) is an arc of C other than (x,y). We also show that if G is arc-disjoint, then G is a fuzzy forest iff there is at most one arc-disjoint strongest path between any two nodes of G. We then turn our attention to the cycle rank of a graph. We show that if G is arc-disjoint and G=C₁∪...∪C_n, where the C _i are cycles, then m(C₁∪...∪C_n)=m(C₁)...m(C_n )=n, where m(·) denotes the cycle rank of a graph. We also show that if G is a cycle vector, then G is arc-disjoint iff m(G)=the number of cycles of G