Fuzzy fractional factors in fuzzy graphs

Abstract

Graph fractional factor theory plays a crucial role in data transmission and network flow existence analysis, and has become one of the hot research branches of graph theory. This paper introduces fuzzy fractional factor in fuzzy graph setting, and an algorithm-based proof of its necessary and sufficient condition is given. The transformation operation is introduced to show that any two fuzzy fractional f -factors can be converted between each other, and the characteristic of maximum fuzzy fractional factor through increasing walk is determined. Finally, toughness in fuzzy graph setting is introduced, and preliminary toughness bound for fuzzy fractional ι-factor is presented, where ι μ e μ e = min { ( ) ( ) > 0}

Fuzzy fractional factors in fuzzy graphs-II

“…The fuzzy factor (FF) and FFF in FG setting are suggested by Gao et al [9] which are formulated by the following definitions respectively. Definition 1.…”

“…Definition 1. (Gao et al [9]) Let G = (V, A, B) be a FG with MFs µ A and µ B and g, f :→ R + ∪ {0} be two functions satisfying g(v) ≤ f (v) for any v ∈ V . We say a FG G admits a fuzzy (g, f )-factor if there is a fuzzy indicator function (FIF) h : E → {0, µ B } satisfying g(v) ≤ ∑ e∼v h(e) ≤ f (v) for any v ∈ V , where e ∼ v implies edge e is incident with vertex v. The fuzzy (g, f )-factor w.r.t.…”

“…In this part, new notations raised in Gao et al [9] are modified to PTFG setting, and all graphs described in this subsection are considered to be PTFG with MFs (µ A , ν A , ξ A , µ B , ν B , ξ B ) such that (µ A , ν A , ξ A ) are PTFSs defined on V and (µ B , ν B , ξ B ) are PTFSs defined on V ×V .…”

“…Due to the similarity of the proof method to Gao et al [9], the detailed process of the above proof has been omitted. However, we still provide some overview of the overall idea.…”

“…Furthermore, the data transmission on the fuzzy graph needs to be characterized by the fractional factor in the corresponding fuzzy graph. To this purpose, Gao et al [9] first introduced the fuzzy fractional factor (FFF) in fuzzy graph (FG) setting, and the necessary and sufficient condition and correlated theoretical properties for the existence of FFFs are determined. However, from the prospect of representation theory, there are obvious flaws in FGs to characterize the uncertainty of graphs.…”