In data storage, graphs are frequently utilized to represent structured or irregular data. In the computer network, the graph structure stores the topological characteristics of the network and is the basis for the analysis of the entire network connectivity, data flow and service flow. When the data to be represented has uncertainties, or the sites and channels of the network itself have uncertainties, then the membership functions are employed to describe such uncertainties, thereby transforming the original vertex set into a fuzzy set, and the edge set is a binary fuzzy relation set. In this way, the entire graph is termed as a fuzzy graph.
The study on fuzzy graphs is the focused topic fuzzy mathematics and graph theory in recent years, involving two fields of mathematical theory and computer algorithm. Akram et al. [1] discussed the threshold graphs involving Pythagorean fuzzy information. Yuan and Wang [2] raised a spectral rotation system with fuzzy anchor. Raut and Pal [3] studied the coloring number and perfectness of fuzzy graphs. Perumal [4] proposed a file clustering approach in terms of fuzzy association rule generation trick. Ullah et al. [5] gave the theoretical analysis on complex
In computer networks, the feasibility of data transmission can be modeled by network flow which is characterized by the existence of fractional flows. Formally, the fractional flow is represented by a fractional factor in a specific graph framework, and in this way, the existence of the fractional factor in a particular setting can be used to describe the characteristics of data transmission in the network.
In a real scenario, realistic factors such as bandwidth, throughput, uplink and downlink capacity, delay, and model capacity need to be considered, resulting in a large number of uncertain qualities in the network. These uncertainties can be characterized by the uncertainty of the sites and the uncertainty of the channels, which are stated as follows.
Site uncertainty: Each site has a different storage and model capacity, resulting in differences in computing ability and throughput. Hence, it inevitably leads to site uncertainty during data transmission. In addition, human factors, such as geolocation of the site, scientific staff configuration, and leadership management capabilities, will also bring uncertainties to sites.
Channel uncertainty: The bandwidth, delay, quality of inlets at both site ends, and link capacity of the channel bring the uncertainty of data transmission. Human factors, such as the degree of connection and cooperation between the stations at both ends of the channel, the response speed and work efficiency of maintenance personnel, also contribute to channels uncertainties.
In order to describe the uncertainty of sites and channels in the network graph, the entire network is represented by a fuzzy graph, and the uncertainty of sites and channels is described by the membership function (MF) of vertices and edges. 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.
It is well acknowledged that an important difference between fuzzy sets and universe sets lies in that MF does not satisfy the law of complementarity. For example: let
To solve the aforementioned problem, multiple MFs are often defined to describe the uncertainty from multiple different angles. Frequently used fuzzy sets are summarized as follows.
Intuitionistic fuzzy set (IFS):
Pythagorean fuzzy set (PFS):
Picture fuzzy set (PTFS):
It is noteworthy that in IFS and PFS,
When it comes to FG setting, a single vertex MF cannot fully describe the vertex uncertainty, and a single edge MF cannot globally measure edge uncertainty. Hence, Parvathi and Karunambigai [10] introduced intuitionistic fuzzy graph (IFG), Naz et al. [11] raised the idea of Pythagorean fuzzy graph (PFG), and Zuo et al. [12] defined picture fuzzy graph (PTFG), where multiple vertex (edge) MFs describe the uncertainty of vertices (edges) from different angles. More fractional studies can be referred to Ata and Kıymaz [13], [14] and [15], Ata [16], Dokuyucu et al. [17] and Veeresha et al. [18]. This motivates us to extend the corresponding FFF to IFG, PFG and PTFG.
In this contribution, the relevant concepts for FFF in IFG, PFG and PTFG settings are defined respectively which included sign-alternating walk, transformation operation and increasing walk, and the corresponding necessary and sufficient conditions are determined by means of alternating path. Moreover, the features of the maximum fuzzy fraction factor (MFFF) are characterized.
The organization of reminder parts are arranged as follows. Firstly, the notations and terminologies are manifested in the next section. Secondly, the new concepts related to FFF in associated settings are defined. Thirdly, we analyze the characterizes of FFF from a theoretical prospective. Finally, two open problems are raised at the end of article.
The aim of this section is to present the terminologies in fuzzy graph setting.
Let
A fuzzy graph
A FG
A fuzzy graph
The fuzzy factor (FF) and FFF in FG setting are suggested by Gao et al. [9] which are formulated by the following definitions respectively.
(Gao et al. [9]) Let
(Gao et al. [9]) Let
Now, we raise the concepts of FF and FFF in IFGs.
(FF in IFG) Let
(FFF in IFG) Let
Analogously, the FF and FFF in PFG can be defined using the same fashion. For completeness, we list the specific definitions below.
(FF in PFG) Let
(FFF in PFG) Let
By extending the number of MFs to three, and using the same trick, the definitions of FF and FFF in PTFGs are obtained.
(FF in PTFG) Let
(FFF in PTFG) Let
In order to obtain the corresponding theoretical results, we need to introduce corresponding concepts on the three types of fuzzy graphs.
In this part, we expend terminologies raised in Gao et al. [9] to IFG and PFG setting, and all graphs described in this subsection are considered to be IFG or PFG with MF (
Let
(Sign-alternating walk) A sign-alternating walk (SAW) (each edge is allowed to appear at most twice) of IFG (resp. PFG)
(Transformation operation) Let If the signs of Δ If the signs of Δ
After the transformation operation, the parallel edges
(Maximum fuzzy fractional factor) Let
(Increasing walk) The increasing walk its edge values under its edge values under
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 (
Let
(Sign-alternating walk) A sign-alternating walk (SAW) (each edge is allowed to appear at most twice) of picture fuzzy graph
(Transformation operation) Let For Δ For Δ For Δ
After the transformation operation, the parallel edges
(Maximum fuzzy fractional factor) Let
(Increasing walk) The increasing walk
In this section, we aim to extend the results on the FFF of FGs presented in Gao et al. [9] to IFG, PFG, and PTFG settings.
The tricks to prove the following theorems are analogue to Gao et al. [9], and we will not repeat it here.
Obviously, Theorem 1 has the following equivalent version.
Let fuzzy fractional (
Similarly, the technologies to prove theorems in this section are analogue to Gao et al. [9], we skip the detailed proofs here.
Obviously, Theorem 5 has the following equivalent version.
Let fuzzy fractional (
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. The proof of the results is algorithmic and can be extended to Algorithm 1 in [9]. The key technology is to start with augmenting path and sign alternating walk, continuously modifying the value of the fractional indicator functions, in order to achieve the effect of transformation. Although multiple MFs appear in the settings of this article, each membership function can be treated similarly to [9] and combined to obtain the results.
In this contribution, we proposed the new concepts of FFF in IFG, PFG and PTFG, respectively. Moreover, the theoretical results involving the necessary and sufficient condition of existence of FFF, and the characteristics of MFFF were determined by means of structure tricks.
Although the concepts and properties of the three types of FGs have been confirmed in this paper, the properties of the FFF in specific FG settings are still open. We believe that the following questions can be the subject of future research.
This paper is a pure theoretical work and has been developed without any data.
H.Z.-Conceptualization, Methodology, Formal analysis, Writing - Original Draft. J.G.-Investigation, Writing-Original Draft. W.G.-Writing, Supervision, Original Draft, Conceptualization, Methodology, Formal analysis, The authors have worked equally when writing this paper. All authors read and approved the final manuscript.
Not applicable.
The authors hereby declare that there is no conflict of interests regarding the publication of this paper.
Many thanks to the reviewers for their constructive comments on revisions to the article. The research is partially supported by NSFC (no. 12161094).
The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.