Complete Fuzzy Graphs Research Articles

A characterization for fuzzy strong cut vertices and fuzzy strong cut edges

Cut vertices and cut edges are valuable for analyzing connectivity problems in classical graph theory. However, they cannot deal with certain fuzzy problems. In order to solve this problem, this paper introduces the definitions of fuzzy strong cut vertices and fuzzy strong cut edges, and characterizes fuzzy strong cut vertices and fuzzy strong cut edges in fuzzy trees, complete fuzzy graphs, and fuzzy cycles. Finally, practical applications verify the effectiveness of the theory in network stability analysis.

Aggregation of fuzzy graphs

Our study is centered on the aggregation of fuzzy graphs, looking for conditions under which the aggregation process yields another fuzzy graph. We conduct an in-depth analysis of the preservation of several important properties and structures inherent to fuzzy graphs, like paths, cycles, or bridges. In addition we obtain appropriate criteria for when the aggregation of complete fuzzy graphs is again a complete fuzzy graph.

First and second K Banhatti indices of a fuzzy graph applied to decision making

Topological index of a graph is a numerical representation linked to the graph or the molecular structure associated with the graph, shedding light on its topological and structural properties. Different topological indices showcase unique attributes rooted in their degree, spectrum, and distance characteristics. Among these, the first and second K Banhatti indices, introduced by Kulli, involving both vertex and edge degrees, have garnered significant scholarly interest, prompting extensive discussions in graph theory. Within this research paper, a thorough exploration delves into the attributes surrounding the fuzzified first and second K Banhatti indices. This exploration is directed towards several bounds of the index and index of various structures like cycles, complete fuzzy graphs, complete bipartite graphs and stars. Relation between a fuzzy graph and its partial subgraphs, fuzzy graph and its complement graph in connection with the indices is also found. Additionally, the research paper investigates the first and second K Banhatti indices concerning graph operations such as the corona product and coalescence. Additionally, the paper outlines an algorithm designed to accurately compute the first and second K Banhatti indices and proposes an application in decision-making.

Geodetic domination integrity in fuzzy graphs

Let N = (V, E) be a simple graph and let X be a subset of V (N). If every node not in X lies on a geodesic path between two nodes from X then it is called a geodetic set. The geodetic number g (N) is the minimum cardinality of such set X. The subset X is called a dominating set if every node not in X has at least one neighbour in X. The minimum number of nodes of a dominating set is known as domination number γ (N). If the subset X is a geodetic set as well as a dominating set then it is called a geodetic dominating set. The minimum cardinality of a geodetic dominating set is known as geodetic domination number γg (N). The geodetic domination integrity of N is defined to be DIg (N) = min {|X| + m (N - X) : X is a geodetic dominating set of N}, where m (N - X) denotes the order of the largest component of N - X. Uncertain networks can be modelled using fuzzy graphs. In a graph, each vertex and each edge are equally significant. However, in fuzzy graphs, each vertex and each edge is important in terms of fuzziness in their own right. In this study, the concepts of geodetic dominating sets in fuzzy graphs and geodetic domination number are defined and bounds are obtained. Moreover, the vulnerability parameter Geodetic domination integrity is introduced in fuzzy graphs. Further, the geodetic domination integrity for complete fuzzy graphs, complete bipartite fuzzy graphs, Cartesian product of two strong fuzzy graphs and bounds are also discussed. The applications of this parameter are applied to a telecommunication network system model to identify the key persons in the system and applied in a fuzzy social network to find the most influential group within the network.

Brain Network Analysis Through Span Integrity of Fuzzy Graphs

Spanness of fuzzy graph is introduced. By spanness, a new vulnerability parameter, span integrity is defined in fuzzy graph. The span integrity values are found for path, cycle, complete fuzzy graph, complete bipartite fuzzy graphs. Path and cycle with node strength sequence are discussed. Brain network is modeled as a fuzzy graph and Span integrity is applied to the brain network. Span integrity of fuzzy brain network is calculated for before and after meditation models. The results are compared and the improvement in the stability of the brain network is shown.

Domination number of complete restrained fuzzy graphs

This work is concerned with the restrained complete domination number and triple connected domination number of fuzzy graphs. Some basic definitions and needful results are given with an example. The necessary and sufficient conditions for the fuzzy graph to be a complete restrained domination set is formulated and proved. Also the relation between complete restrained domination set and n-dominated set is illustrated. Finally, triple connected domination number of a restrained complete fuzzy graph is provided.

Characterization of a Vertex Colouring of a Double Layered Complete Fuzzy Graph Using ? – CUT

Abstract: In this paper we defined a new fuzzy graph named Double layered complete fuzzy graph. (DLCFG). The double layered complete fuzzy graph gives a 3-D structure. Further we introduced vertex colouring of the double layered complete fuzzy graph using α-cut. Keywords: Graph theory, Fuzzy graph, colouring of graphs, double layered fuzzy graph, complete fuzzy graph, alpha cut, colouring of double layered fuzzy graph, colouring of double layered complete fuzzy using alpha cut.

ON THE COMPLETE PRODUCT OF FUZZY GRAPHS

Strong fuzzy graph and complete fuzzy graph are two special kinds of fuzzy graphs in which each edge membership value equals the least of its vertex membership values incidental to the edge. In this research study, we obtain some interesting results on the complete product of a pair of strong fuzzy graphs as well as the complete product of a pair of complete fuzzy graphs. In precise, we prove that the complete product of two strong fuzzy graphs is again a strong fuzzy graph and the complete product of two complete fuzzy graphs is again a complete fuzzy graph. Also, we discuss the conditions under which the property of regularity will be mutually transmitted between the complete product of two fuzzy graphs and one of its factor fuzzy graphs

Set Domination in Fuzzy Graphs Using Strong Arcs

Set domination in fuzzy graphs is very useful for solving traffic problems during communication in computer networks and travelling networks. In this article, the concept of set domination in fuzzy graphs using strong arcs is introduced. The strong set domination number of complete fuzzy graph and complete bipartite fuzzy graph is determined. It is obtained the properties of the new parameter and related it to some other known domination parameters of fuzzy graphs. An upper bound for the strong set domination number of fuzzy graphs is also obtained.

Strong domination integrity in graphs and fuzzy graphs

Let H = (V, E) be a graph and xy ∈ E (H). Then x strongly dominates y if deg(x) ⩾ deg(y). A subset S of V is said to be a strong dominating set if every node y ∈ V – S is strongly dominated by some node x in H and is denoted by sd-set. The strong domination number γs (H) is the minimum cardinality of a strong dominating set. In this paper, we introduce a new vulnerability parameter called strong domination integrity in graphs. Strong domination integrity of some families of graphs are determined and its bounds are also obtained. The proposed parameter is applied in water distribution network system to identify the influential group of nodes within the network. Fuzzy graphs can be used to model uncertain networks. By using membership values of strong arcs, strong domination integrity is extended to fuzzy graphs as a new vulnerability parameter. In this study, we investigate the strong domination integrity for complete bipartite fuzzy graphs, complete fuzzy graphs and bounds are also derived. Some basic results and theorems are obtained. This vulnerability parameter is also applied in the transportation network systems.

Estimation of most effected cycles and busiest network route based on complexity function of graph in fuzzy environment.

Connectivity and strength has a major role in the field of network connecting with real world life. Complexity function is one of these parameter which has manifold number of applications in molecular chemistry and the theory of network. Firstly, this paper introduces the thought of complexity function of fuzzy graph with its properties. Second, based on the highest and lowest load on a network system, the boundaries of complexity function of different types of fuzzy graphs are established. Third, the behavior of complexity function in fuzzy cycle, fuzzy tree and complete fuzzy graph are discussed with their properties. Fourth, applications of these thoughts are bestowed to identify the most effected COVID-19 cycles between some communicated countries using the concept of complexity function of fuzzy graph. Also the selection of the busiest network stations and connected internet paths can be done using the same concept in a graphical wireless network system.

Wiener Index of Some Operations on Fuzzy Graphs

In this paper, the concept of the Wiener index in some operations on fuzzy graphs was introduced and investigated. The bound of W(G) of some operations on fuzzy graphs are obtained like union, join, Cartesian product, composition, complete fuzzy graph, and bipartite complete fuzzy graph.

Fuzzy Detour Convexity and Fuzzy Detour Covering in Fuzzy Graphs

A path P connecting a pair of vertices in a connected fuzzy graph is called a fuzzy detour, if its μ - length is maximum among all the feasible paths between them. In this paper we establish the notion of fuzzy detour convex sets, fuzzy detour covering, fuzzy detour basis, fuzzy detour number, fuzzy detour blocks and investigate some of their properties. It has been proved that, for a complete fuzzy graph G, the set of any pair of vertices in G is a fuzzy detour covering. A necessary and sufficient condition for a complete fuzzy graph to become a fuzzy detour block is also established. It has been proved that for a fuzzy tree there exists a nested chain of sets, where each set is a fuzzy detour convex. Application of fuzzy detour covering and fuzzy detour basis is also presented.

First Zagreb index on a fuzzy graph and its application

The Zagreb index (ZI) is a very important graph parameter and it is extensively used in molecular chemistry, spectral graph theory, network theory and several fields of mathematics and chemistry. In this article, the first ZI is studied for several fuzzy graphs like path, cycle, star, fuzzy subgraph, etc. and presented an ample number of results. Also, it is established that the complete fuzzy graph has maximal first ZI among n-vertex fuzzy graphs. Some bounds of first ZI are discussed for Cartesian product, composition, union and join of two fuzzy graphs. An algorithm has been designed to calculate the first ZI of a fuzzy graph. At the end of the article, a multi-criteria decision making (MCDM) method is provided using the first ZI of a fuzzy graph to find the best employee in a company. Also a comparison is provided among related indices on the result of application and shown that our method gives better results.

SIFAT ISOMORFIK PADA OPERASI TENSOR, BINTANG, CARTESIUS, DAN MODULAR DUA GRAF FUZZY

This article discusses about some isomorphic properties of tensor, star, Cartesius, and modular product operations of two fuzzy graphs. The results of research are the tensor product of two fuzzy graphs is isomorphic, and if and are complete fuzzy graphs then , , and .
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Properties of cartesian multiplication operations in complete fuzzy graphs, effective fuzzy graphs and complement fuzzy graphs

This study aims to investigate the properties of cartesian multiplication operations on fuzzy graphs. The properties investigated are cartesian multiplication on complete fuzzy graph, effective fuzzy graph and complement fuzzy graph. The method used in this study is a theoretical research method. The results showed that the Cartesian multiplication of two complete fuzzy graphs is not a complete fuzzy graph. The Cartesian multiplication of the two effective fuzzy graphs is the effective fuzzy graph, while for the Cartesian multiplication of the two fuzzy complement graphs, it is not the same as the Cartesian multiplication of the two effective fuzzy graphs.

Cyclic Connectivity Index of Fuzzy Graphs

A parameter is a numerical or other measurable factor whose values characterize a system. Connectivity parameters have indispensable role in the analysis of connectivity of networks. Strength of a cycle in an unweighted graph is always one. But, in a fuzzy graph, strengths of cycles may vary even for a given pair of vertices. Cyclic reachability is a property that determines the overall connectedness of a network. This article introduces two connectivity parameters namely, cyclic connectivity index (CCI) and average CCI (ACCI) of fuzzy graphs, which can be used to represent the cyclic reachability. CCI of fuzzy graph theoretic structures, such as trees, blocks, $\theta$ -fuzzy graphs, and complete fuzzy graphs, are discussed. Vertices of a fuzzy graph are classified into three categories in terms of ACCI and their characterizations are obtained. Three algorithms are proposed. One of them is to help find CCI and ACCI of a given fuzzy graph. Another helps to identify the nature of vertices and the third to enhance the existing ACCI of a fuzzy graph. Also, future directions in the study of CCI and ACCI are proposed.

Strong Fuzzy Chromatic Polynomial (SFCP) of Fuzzy Graphs and Some Fuzzy Graph Structures with Applications

In fuzzy graph theory, strong arcs have separate importance. Assign different colors to the end nodes of strong arcs in the fuzzy graph is strong coloring. Strong coloring plays an important role in solving real-life problems that involve networks. In this work, we introduce the new concept, called strong fuzzy chromatic polynomial (SFCP) of a fuzzy graph based on strong coloring. The SFCP of a fuzzy graph counts the number of <i>k</i>-strong colorings of a fuzzy graph with <i>k</i> colors. The existing methods for determining the chromatic polynomial of the crisp graph are used to obtain SFCP of a fuzzy graph. We establish the necessary and sufficient condition for SFCP of a fuzzy graph to be the chromatic polynomial of its underlying crisp graph. Further, we study SFCP of some fuzzy graph structures, namely strong fuzzy graphs, complete fuzzy graphs, fuzzy cycles, and fuzzy trees. Besides, we obtain relations between SFCP and fuzzy chromatic polynomial of strong fuzzy graphs, complete fuzzy graphs, and fuzzy cycles. Finally, we present dual applications of the proposed work in the traffic flow problem. Once SFCP of a fuzzy graph is obtained, the proposed approach is simple enough and shortcut technique to solve strong coloring problems without using coloring algorithms.

Fuzzy Chromatic Polynomial of Fuzzy Graphs with Crisp and Fuzzy Vertices Using α-Cuts

Coloring of fuzzy graphs has many real life applications in combinatorial optimization problems like traffic light system, exam scheduling, register allocation, etc. In this paper, the concept of fuzzy chromatic polynomial of fuzzy graph is introduced and defined based on α-cuts of fuzzy graph. Two different types of fuzziness to fuzzy graph are considered in the paper. The first type was fuzzy graph with crisp vertex set and fuzzy edge set and the second type was fuzzy graph with fuzzy vertex set and fuzzy edge set. Depending on this, the fuzzy chromatic polynomials for some fuzzy graphs are discussed. Some interesting remarks on fuzzy chromatic polynomial of fuzzy graphs have been derived. Further, some results related to the concept are proved. Lastly, fuzzy chromatic polynomials for complete fuzzy graphs and fuzzy cycles are studied and some results are obtained.

Geodesic Iteration Number of g-contour of a Fuzzy Graph

Purpose: In this paper, we intent to introduce the concept of g-contour nodes in fuzzy graphs and demonstrate an application of g-contour nodes in Land line telecommunication Network. Design/approach: The concept of g-contour nodes in Fuzzy graphs is explained and is supported by several examples. The geodesic iteration number of g-contour nodes in fuzzy graphs is obtained. An application of geodesic iteration number of g-contour nodes for determining the central persons in the land-line telecommunication network (LTN) is demonstrated, by means of which churning in the land-line telecommunication system can be reduced by providing a step-wise method for canvassing each and every customer in the network. Findings: It is proved that every extreme node of G is a g-contour node of G but not conversely. The g-contour of g-self-centred fuzzy graphs and of fuzzy trees is found to be a geodesic cover. The geodesic iteration number of the g-contour of a fuzzy graph is obtained. The geodesic iteration number of complete fuzzy graphs and of fuzzy trees is proved to coincide with the geodesic iteration number of their g-contours.