Vertex-strength of fuzzy graphs

The chromatic sum of a graph G, ∑ (G), is introduced in the dissertation of Kubicka [3]. It is defined as the smallest possible total over all vertices that can occur among all colorings of G using natural numbers for the colors. It is known that computing the chromatic sum of an arbitrary graph is an NP-complete problem. The vertex-strength of the graph G, denoted by s(G), is the smallest integer s such that ∑ (G) is attained using colors {1,2, . . . ,s}. In this article, we generalize these concepts to fuzzy graphs and wish to bound chromatic sum of a fuzzy graph with the e strong edges inG. We review some of the definitions of fuzzy graphs as in [2, 6, 7] and introduce some new notations. Let X be a nonempty set and E the collection of all two-element subsets of X . A fuzzy set γ on X is a mapping γ : X → [0,1]. Given α∈ (0,1], the α-cut of γ is defined by γα = {x ∈ X | γ(x)≥ α}. The support and height of γ are defined by suppγ = {x ∈ X | γ(x) > 0} and h(γ)=max{γ(x) | x ∈ X}, respectively. Fuzzy intersection of two fuzzy sets γ1 and γ2 is denoted by γ1∧ γ2 =min{γ1,γ2}. Let X be a finite nonempty set. The triple G = (X ,σ ,μ) is called a fuzzy graph on X where σ and μ are fuzzy sets on X and E, respectively, such that μ({x, y})≤min{σ(x), σ(y)} for all x, y ∈ X . Hereafter, we use μ(xy) for μ({x, y}). The fuzzy graphG′=(X ,σ ′,μ′) is called a fuzzy subgraph of G if for each two elements x, y ∈ X , we have σ ′(x) ≤ σ(x) and μ′(xy)≤ μ(xy). The fuzzy graph G= (X ,σ ,μ) is called connected if for every two elements x, y ∈ X , there exists a sequence of elements x0,x1, . . . ,xm such that x0 = x, xm = y and μ(xixi+1) > 0 (0≤ i≤m− 1).