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A fuzzy graph G is stated to have a set of trees as its tree cover if all the vertices of G are in their union. The maximum weight tree in the tree cover is assumed to be the cost of a tree cover for a fuzzy graph. For an integer β > 0 , finding a set of trees to cover all the vertices of a graph with minimum cost and at most β number of spanning trees is known as the β -tree cover problem. Combining the tree-covering concept and facility location problem in a fuzzy environment for solving critical real-life problems in the recent era is a more fruitful approach. This issue strongly inspires us to develop a model with a practical algorithm. This paper provides an algorithm and complexity analysis to determine the number of rooted trees s covering the given fuzzy graph. In addition, a model is constructed with three optimization programming problems in the facility location problem and a tree covering fuzzy graphs. The model includes two types of the facility location problem, simultaneously addressing a variable covering radius and a fixed covering radius. A numerical example is provided to further describe the model, then, in the application part of the paper, the proposed model is applied to solve the real-life problem of maximizing demand saturation by minimizing the number of small denominations in the Indian banking system. This problem involves the data input of different indicators in the banking system along with details of the denominations of banknotes. [ABSTRACT FROM AUTHOR]

In decision support tasks, one often has to deal with uncertainty due to fuzzy judgments of the decision maker or the expert. This paper proposes methods that allow you to rank the alternatives based on fuzzy evaluations. This is achieved by using fuzzy weighted summation, fuzzy implication, a computation graph showing the criteria, and a fuzzy dominance graph showing the alternatives. If the criteria have equal importance, then fuzzy graphs corresponding to the dominance of each of the criteria are used. An algorithm that is used for both the transition from fuzzy dominance graphs and the ranking of alternatives is proposed. This algorithm is based on the idea of constructing Kemeny medians or other concordant rankings at a given confidence level in the existence of corresponding arcs. Computational experiments have shown the performance of these approaches. It is reasonable to apply them in problems that require complex expert evaluations with a large number of alternatives and criteria. [ABSTRACT FROM AUTHOR]

In this paper, we consider the adjacent vertex distinguishing proper edge coloring (for short, AVDPEC) and the adjacent vertex distinguishing total coloring (for short, AVDTC) of a fuzzy graph. Firstly, this paper describes the development process, the application areas, and the existing review research of fuzzy graphs and adjacent vertex distinguishing coloring of crisp graphs. Secondly, we briefly introduce the coloring theory of crisp graphs and the related theoretical basis of fuzzy graphs, and add some new classes of fuzzy graphs. Then, based on the α -cuts of fuzzy graphs and distance functions, we give two definitions of the AVDPEC of fuzzy graphs, respectively. A lower bound on the chromatic number of the AVDPEC of a fuzzy graph is obtained. With examples, we show that some results of the AVDPEC of a crisp graph do not carry over to our set up; the adjacent vertex distinguishing chromatic number of the fuzzy graph is different from the general chromatic number of a fuzzy graph. We also give a simple algorithm to construct a (d , f) -extended AVDPEC for fuzzy graphs. After that, in a similar way, two definitions of the AVDTC of fuzzy graphs are discussed. Finally, the future research directions of distinguishing coloring of fuzzy graphs are given. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the double domination in vague graphs, and it is a contribution to the Special Issue "Advances in graph theory and Symmetry/Asymmetry" of Symmetry. Symmetry is one of the most important criteria that illustrate the structure and properties of fuzzy graphs. It has many applications in dominating sets and helps find a suitable place for construction. Vague graphs (VGs), which are a family of fuzzy graphs (FGs), are a well-organized and useful tool for capturing and resolving a range of real-world scenarios involving ambiguous data. In the graph theory, a dominating set (DS) for a graph G * = (X , E) is a subset D of the vertices X so that every vertex which is not in D is adjacent to at least one member of D. The subject of energy in graph theory is one of the most attractive topics serving a very important role in biological and chemical sciences. Hence, in this work, we express the notion of energy on a dominating vague graph (DVG) and also use the concept of energy in modeling problems related to DVGs. Moreover, we introduce a new notion of a double dominating vague graph (DDVG) and provide some examples to explain various concepts introduced. Finally, we present an application of energy on DVGs. [ABSTRACT FROM AUTHOR]

Consider a simple connected fuzzy graph (FG) G and consider an ordered fuzzy subset H = {(u1, σ(u1)), (u2, σ(u2)), ...(uk, σ(uk))}, |H| ≥ 2 of a fuzzy graph; then, the representation of σ − H is an ordered k-tuple with regard to H of G. If any two elements of σ − H do not have any distinct representation with regard to H, then this subset is called a fuzzy resolving set (FRS) and the smallest cardinality of this set is known as a fuzzy resolving number (FRN) and it is denoted by Fr(G). Similarly, consider a subset S such that for any u∈S, ∃v∈V − S, then S is called a fuzzy dominating set only if u is a strong arc. Now, again consider a subset F which is both a resolving and dominating set, then it is called a fuzzy resolving domination set (FRDS) and the smallest cardinality of this set is known as the fuzzy resolving domination number (FRDN) and it is denoted by Fγr(G). We have defined a few basic properties and theorems based on this FRDN and also developed an application for social network connection. Moreover, a few related statements and illustrations are discussed in order to strengthen the concept. [ABSTRACT FROM AUTHOR]

We focus on fuzzy graphs with crisp vertex sets and fuzzy edge sets. This paper introduces a new concept of chromatic number (crisp) for a fuzzy graph G ˜ (V , E ˜) . Moreover, we define the operations of cap, join, difference, ring sum, direct product, semiproduct, strong product, and Cartesian product of fuzzy graphs. Furthermore, the exact value or the upper boundary of the chromatic number of these fuzzy graphs is obtained based on the α -cuts of G ˜ . Finally, two applications of the chromatic number to solve the timetabling problem and the traffic light problem are analyzed. [ABSTRACT FROM AUTHOR]

The advent of fuzzy sets, and consequently fuzzy graphs, has solved many problems in ambiguous and uncertain contexts. It is interesting and necessary to study the Wiener index in a cubic fuzzy graph that employs both fuzzy membership and interval-valued fuzzy membership at the same time. In this paper, the Wiener index in a cubic fuzzy graph is introduced as a cubic fuzzy number and some related results are described. The comparison between connectivity index and Wiener index, changes in Wiener index through deleting a node or an edge, and determining the Wiener index in some specific cubic fuzzy graphs have been the other topics studied in this research. In addition, the Wiener index is determined by mentioning concepts of the saturated cubic fuzzy cycle. In this review, the Wiener index is shown as a combination of classical and interval numbers. The results indicate that when some vertices are removed, the Wiener index may change. However, this change will not be exclusively related to both values. Finally, an application of the Wiener index is presented in the study of the properties of some monomer molecules. [ABSTRACT FROM AUTHOR]

Vague graphs (VGs), belonging to the fuzzy graphs (FGs) family, have good capabilities when faced with problems that cannot be expressed by FGs. The notion of a VG is a new mathematical attitude to model the ambiguity and uncertainty in decision-making issues. A vague fuzzy graph structure (VFGS) is the generalization of the VG. It is a powerful and useful tool to find the influential person in various relations. VFGSs can deal with the uncertainty associated with the inconsistent and indeterminate information of any real-world problems where fuzzy graphs may fail to reveal satisfactory results. Moreover, VGSs are very useful tools for the study of different domains of computer science such as networking, social systems, and other issues such as bioscience and medical science. The subject of energy in graph theory is one of the most attractive topics that is very important in biological and chemical sciences. Hence, in this work, we extend the notion of energy of a VG to the energy of a VFGS and also use the concept of energy in modeling problems related to VFGS. Actually, our purpose is to develop a notion of VFGS and investigate energy and Laplacian energy (LE) on this graph. We define the adjacency matrix (AM) concept, energy, and LE of a VFGS. Finally, we present three applications of the energy in decision-making problems. [ABSTRACT FROM AUTHOR]

Connectivity parameters have an important role in the study of communication networks. Wiener index is such a parameter with several applications in networking, facility location, cryptology, chemistry, and molecular biology, etc. In this paper, we show two notes related to the Wiener index of a fuzzy graph. First, we argue that Theorem 3.10 in the paper "Wiener index of a fuzzy graph and application to illegal immigration networks, Fuzzy Sets and Syst. 384 (2020) 132–147" is not correct. We give a correct statement of Theorem 3.10. Second, by using a new operator on matrix, we propose a simple and polynomial-time algorithm to compute the Wiener index of a fuzzy graph. [ABSTRACT FROM AUTHOR]

Broadcasting problems in graph theory play a significant role in solving many complicated physical problems. However, in real life there are many vague situations that sometimes cannot be modeled using usual graphs. Consequently, the concept of a fuzzy graph G F : (V , σ , μ) has been introduced to deal with such problems. In this study, we are interested in defining the notion of dominating broadcasts in fuzzy graphs. We also show that, in a connected fuzzy graph containing more than one element in σ * , a dominating broadcast always exists, where σ * is { v ∈ V | σ (v) > 0 } . In addition, we investigate the relationship between broadcast domination numbers, radii, and domination numbers in a fuzzy graph as follows; γ b (G F) ≤ min { r (G F) , γ (G F) } , where γ b (G F) is the broadcast domination number, r (G F) is the radius, and γ (G F) is domination numbers in fuzzy graph G F , with | σ * | > 1 . [ABSTRACT FROM AUTHOR]

A new domination parameter in a fuzzy digraph is proposed to espouse a contribution in the domain of domination in a fuzzy graph and a directed graph. Let G D * = V , A be a directed simple graph, where V is a finite nonempty set and A = x , y : x , y ∈ V , x ≠ y . A fuzzy digraph G D = σ D , μ D is a pair of two functions σ D : V → 0 , 1 and μ D : A → 0 , 1 , such that μ D x , y ≤ σ D x ∧ σ D y , where x , y ∈ V . An edge μ D x , y of a fuzzy digraph is called an effective edge if μ D x , y = σ D x ∧ σ D y . Let x , y ∈ V . The vertex σ D x dominates σ D y in G D if μ D x , y is an effective edge. Let S ⊆ V , u ∈ V \ S , and v ∈ S . A subset σ D S ⊆ σ D is a dominating set of G D if, for every σ D u ∈ σ D \ σ D S , there exists σ D v ∈ σ D S , such that σ D v dominates σ D u . The minimum dominating set of a fuzzy digraph G D is called the domination number of a fuzzy digraph and is denoted by γ G D . In this paper, the concept of domination in a fuzzy digraph is introduced, the domination number of a fuzzy digraph is characterized, and the domination number of a fuzzy dipath and a fuzzy dicycle is modeled. [ABSTRACT FROM AUTHOR]

In mathematics, distance and similarity are known as dual concepts. However, the concept of similarity is interpreted as fuzzy similarity or T-equivalence relation, where T is a triangular norm (t-norm in brief), when we discuss a fuzzy environment. Dealing with multi-polarity in practical examples with fuzzy data leadsus to introduce a new concept called m-polar T -equivalence relations based on a finitely multivalued t-norm T , and to study the metric behavior of such relations. First, we study the new operators including the m-polar triangular norm T and conorm S as well as m-polar implication I and m-polar negation N , acting on the Cartesian product of [ 0 , 1 ] m-times.Then, using the m-polar negations N , we provide a method to construct a new type of metric spaces, called m-polar S -pseudo-ultrametric, from the m-polar T -equivalences, and reciprocally for constructing m-polar T -equivalences based on the m-polar S -pseudo-ultrametrics. Finally, the link between fuzzy graphs and m-polar S -pseudo-ultrametrics is considered. An algorithm is designed to plot a fuzzy graph based on the m-polar S L -pseudo-ultrametric, where S L is the m-polar Lukasiewicz t-conorm, and is illustrated by a numerical example which verifies our method. [ABSTRACT FROM AUTHOR]

Graph theory is a well-established concept that is widely used in numerous applications such as in biology, chemistry and network analysis. The advancement in the theory of graph has led to the development of a new concept called fuzzy autocatalytic set. In this paper, a fuzzy graph-based chemometrics method, namely, chemometrics fuzzy autocatalytic set (c-FACS) is developed and applied for gelatin authentication. The issue on authenticity of gelatin has become a sensitive issue among some religious communities. Due to the matter, Fourier transform infrared (FTIR) spectra of bovine, porcine and fish gelatins are analyzed using c-FACS to identify their signatures and differences and presented in this paper. The results from the c-FACS analysis showed distinct features of each gelatin, particularly porcine. Furthermore, the new method is faster than principal component analysis (PCA) in identifying the gelatin sources. [ABSTRACT FROM AUTHOR]

Graph models are fundamental in network theory. But normalization of weights are necessary to deal with large size networks like internet. Most of the research works available in the literature have been restricted to an algorithmic perspective alone. Not much have been studied theoretically on connectivity of normalized networks. Fuzzy graph theory answers to most of the problems in this area. Although the concept of connectivity in fuzzy graphs has been widely studied, one cannot find proper generalizations of connectivity parameters of unweighted graphs. Generalizations for some of the existing vertex and edge connectivity parameters in graphs are attempted in this article. New parameters are compared with the old ones and generalized values are calculated for some of the major classes like cycles and trees in fuzzy graphs. The existence of super fuzzy graphs with higher connectivity values are established for both old and new parameters. The new edge connectivity values for some wider classes of fuzzy graphs are also obtained. The generalizations bring substantial improvements in fuzzy graph clustering techniques and allow a smooth theoretical alignment. Apart from these, a new class of fuzzy graphs called generalized t-connected fuzzy graphs are studied. An algorithm for clustering the vertices of a fuzzy graph and an application related to human trafficking are also proposed. [ABSTRACT FROM AUTHOR]

The picture fuzzy set is an efficient mathematical model to deal with uncertain real life problems, in which a intuitionistic fuzzy set may fail to reveal satisfactory results. Picture fuzzy set is an extension of the classical fuzzy set and intuitionistic fuzzy set. It can work very efficiently in uncertain scenarios which involve more answers to these type: yes, no, abstain and refusal. In this paper, we introduce the idea of the picture fuzzy graph based on the picture fuzzy relation. Some types of picture fuzzy graph such as a regular picture fuzzy graph, strong picture fuzzy graph, complete picture fuzzy graph, and complement picture fuzzy graph are introduced and some properties are also described. The idea of an isomorphic picture fuzzy graph is also introduced in this paper. We also define six operations such as Cartesian product, composition, join, direct product, lexicographic and strong product on picture fuzzy graph. Finally, we describe the utility of the picture fuzzy graph and its application in a social network. [ABSTRACT FROM AUTHOR]

Fuzzy graph theory plays an important role in the study of the symmetry and asymmetry properties of fuzzy graphs. With this in mind, in this paper, we introduce new neutrosophic graphs called complex neutrosophic graphs of type 1 (abbr. CNG1). We then present a matrix representation for it and study some properties of this new concept. The concept of CNG1 is an extension of the generalized fuzzy graphs of type 1 (GFG1) and generalized single-valued neutrosophic graphs of type 1 (GSVNG1). The utility of the CNG1 introduced here are applied to a multi-attribute decision making problem related to Internet server selection. [ABSTRACT FROM AUTHOR]