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Recently, vague graph is a highly growing research area as it is the generalisation of the fuzzy graphs. In this paper, we analysed the concepts of edge regular product vague graphs and its properties. The concepts of edge irregular product vague graphs, strongly edge irregular product vague graphs are analysed with properties.

Fuzzy graph theory is applicable on diverse fields where uncertainty plays a significant role like in network analysis, medical diagnosis and health care, decision-making, and optimization, etc. An effective tool in this field is the topological index. These indices provide quantitative measures of molecular structure and connectivity, which are crucial for understanding and predicting physical, chemical, and biological properties of molecules. The Gutman index (GI) is an important topological index. This index is based on degree and distance. In this paper, GI is explored in fuzzy graphs. Some results on GI are found like the relationship of GI between a fuzzy graph and its fuzzy sub-graph and the relationship of GI between a fuzzy graph and its complete fuzzy graph. Some upper and lower bounds are obtained for complete fuzzy graph, bipartite graph, star, double star, etc. A relationship of GI between two isomorphic fuzzy graphs are determined. Human trafficking, a heinous offense and grave breach of human rights involves the coercion, deception, or coercion of people for a range of purposes, including forced work, forced marriage, and forced servitude. Overall, human trafficking is a complex and multifaceted issue that requires coordinated efforts at the local, national, and international levels to prevent, prosecute, and ultimately eradicate this egregious violation of human dignity. At the end of this paper, an application of GI is given to reduce human trafficking in India.

Yang et al. (2013) introduced the concept of generalised bipolar fuzzy graphs in 2013. In this paper, we have introduced certain concepts of covering, matching and paired domination using strong arcs in bipolar fuzzy graphs with suitable examples. We investigated some properties of them. Also, we have calculated strong node covering number, strong independent number and other parameters of complete and complete bipartite bipolar fuzzy graphs.

In this study, the notions of picture fuzzy ϕ-tolerance competition graphs are exhibited adding more uncertainties to the fuzzy ϕ-tolerance competition graphs. Taking ϕas max, min and sum functions three special types of picture fuzzy ϕ-tolerance competition graphs are introduced and few interesting properties of them are studied. The operations of Cartesian product, composition and union between two picture fuzzy digraphs are studied. Also, few hereditary properties of picture fuzzy ϕ-tolerance competition graphs under these operations are illustrated. Two real life applications in railway network and medical science are modeled using picture fuzzy min-tolerance competition graph.

Fuzzy mixed graph (FMG) can be used to model some graphical intercommunication network systems if there are many directed and undirected relations between some vertices. In inverse fuzzy graph, the membership value of edges are greater than or equal to the minimum of the membership value of the corresponding vertices. In inverse fuzzy mixed graph (IFMG), directed and undirected relations exist between some vertices and it can be used to analyze many graphical problems of real life such that the membership values of edges are greater than or equal to the minimum of the membership value of the corresponding pair of vertices. In this article, the concept of IFMG is introduced first with some of its properties. Then some isomorphic properties are studied and complement of IFMG is given. Different types of operations like union, intersection, product and join between two IFMGs are defined and investigated some of their related results. An algorithm of the proposed method is executed to identify some vertices. An application is depicted using the concept IFMG to examine the order of vertices in a social network group according to communication gaps.

Due to the presence of two opposite directional thinking in relationships between countries and communication systems, the systems may not always be balanced. Therefore, the perfectness between countries relations are highly important. It comes from how much they were connected to each other for communication. In this study, first perfectly regular bipolar fuzzy graph is introduced and examined the regularity of nodes. Then, the relationship between the adjacent nodes and their regularity are visualized as a perfectly edge-regular bipolar fuzzy graphs. The totally accurate communication between all connected nodes is explained by introducing completely open neighborhood degree and completely closed neighborhood degree of nodes and edges in a bipolar fuzzy graph. Some algorithms and flowcharts of the proposed methods are given. Finally, two applications of these cogitation are exhibited in two bipolar fuzzy fields. The first one is in international relationships between some countries during cold-war era and the second one is in decision-making between teachers–students communication system for the improvement of teaching.

In this study, the embedding of picture fuzzy graphs on the surface of spheres is introduced. The picture fuzzy genus graph with its genus value and strong (weak) picture fuzzy genus graphs are defined. Some important properties of picture fuzzy genus graphs are discussed. A relation between genus value and degree of planarity of picture fuzzy graph is established. The isomorphism properties of picture fuzzy genus graphs are studied. The Euler polyhedral equation in terms of genus value is established here. Finally, two real-life applications of picture fuzzy genus graph in traffic problem on road network and electronic circuit design on planar sheet are exhibited.

In this paper, adjacency sequence of a vertex, first and second fundamental sequences are defined in a bipolar fuzzy graph with example. Some examples are constructed to show that if Gis a regular bipolar fuzzy graph (BFG), the underlying crisp graph need not be regular and all the vertices need not have the same adjacency sequence. Also it is shown that if Gand its underlying crisp graph are regular, all the vertices need not have the same adjacency sequence. A necessary and sufficient condition is established for a BFG with at most four vertices to be regular using the concept of adjacency sequences. Moreover, some characterizations have been made for a line graph of a regular BFG to be regular, the complement of a regular BFG to be regular, etc.

The idea of inverse fuzzy mixed graph (IFMG) was firstly introduced by Poulik and Ghorai (Granul Comput 7:549–559, 2021) in 2021 generalizing the idea of fuzzy mixed graphs. Here the idea of planarity is introduced in the more generalized setting of IFMGs using the concept of intersecting value. The concept of effective edges and strong planar graphs are given. Also the idea of I-considerable edges and I-considerable numbers have been discussed at the end. Several properties of IFMGs are studied. Finally an application is discussed using these concepts.

In this study, the notions of picture fuzzy tolerance graphs, picture fuzzy interval containment graphs and picture fuzzy ϕ-tolerance graphs are established. Three special types of picture fuzzy tolerance graphs having bounded representations are introduced and studied corresponding properties of them taking ϕas max, min and sum functions. Also, picture fuzzy proper and unit tolerance graphs are established and some related results are investigated. The class of picture fuzzy ϕ-tolerance chaingraphs which is the picture fuzzy ϕ-tolerance graphs of a nested family of picture fuzzy intervals are presented. A real-life application in sports competition is modeled using picture fuzzy min-tolerance graph. Also a comparison is given between picture fuzzy tolerance graphs and intuitionistic fuzzy tolerance graphs.

The quadripartitioned neutrosophic set is defined by dividing the indeterminacy function of the neutrosophic set into the contradiction and ignorance components. Quadripartitioned neutrosophic sets offer an additional means to represent imprecise, uncertain, inconsistent, and incomplete information encountered in real-world scenarios. This research paper introduces the concept of a quadripartitioned neutrosophic competition graph for the first time with its application to graphs. Furthermore, the paper establishes operations for quadripartitioned neutrosophic competition graphs, such as the Cartesian product and direct product. The proposed concepts are exemplified and applied to a quadripartitioned neutrosophic digraph in the context of travel time. Moreover the developed concepts are applied to an optimization techniques on quadripartitioned neutrosophic competition graph with sensitivity analysis, which studies the robustness to the proposed quadripartitioned neutrosophic competition framework.

A graph structure is a useful tool in solving the combinatorial problems in different areas of computer science and computational intelligence systems. In this paper, we present a frame work to handle m-polar fuzzy information by combining the theory of m-polar fuzzy sets with graphs. We introduce the notion of strongly edge irregular and strongly edge totally irregular m-polar fuzzy graphs. Some properties of them are also studied to characterize strongly edge irregular and strongly edge totally irregular m-polar fuzzy graphs.

Recently, bipolar fuzzy graph is a vastly growing research area as it is the generalization of the fuzzy graphs. In this paper, at first the concepts of regular and totally regular product bipolar fuzzy graphs is introduced. Then necessary and sufficient conditions are established under which regular and totally regular product bipolar fuzzy graph becomes equivalent. The notion of product bipolar fuzzy line graph is introduced and investigated some of its properties. A necessary and sufficient condition is given for a product bipolar fuzzy graph to be isomorphic to its corresponding product bipolar fuzzy line graph. It is also examined when an isomorphism between two product bipolar fuzzy graphs follows from an isomorphism of their corresponding fuzzy line graphs.

In many practical applications with a graph structure, there may exist crossing between edges which is not allowed in a crisp planar graph. Considering that graphic structures can be used to describe the crossing of edges in an m-polar fuzzy multi-graph with certain amount of m-polar fuzzy planarity value, first of all the notion of m-polar fuzzy multi-set is introduced. Then m-polar fuzzy multi-graphs, m-polar fuzzy planar graphs and m-polar fuzzy strong edges are defined. Several properties of m-polar fuzzy planar graphs are also studied.

The competition graph is a well-studied topic. But, the fuzzy competition graph has been defined recently and investigated many properties. Only one component is considered for every vertex and edge in a fuzzy competition graph. In an interval-valued m-polar fuzzy graph, each vertex and each edge have mnumber of membership values, all of which are intervals in [0, 1]. So, defining a competition graph for an interval-valued m-polar fuzzy graph is not easy and needs some new ideas. By considering minterval-valued components of the membership value of a vertex or edge, interval-valued m-polar fuzzy competition graph (IVmPFCG) has been defined. Many interesting properties are presented on IVmPFCG. Here, two more generalizations of IVmPF competition graph, namely interval-valued m-polar fuzzy k-competition graph (IVmPFkCG) and interval-valued m-polar fuzzy p-competition graph (IVmPFkCG) are introduced and studied some of their interesting properties. These graphs are directly related to directed IVmPFG. Based on open and closed neighborhoods of vertices, different types of neighborhood graphs are defined and studied for some of their intersecting properties. Also, another type of competition graph, namely q-step interval-valued m-polar fuzzy competition graph, is introduced and investigated some of its exciting properties thoroughly. Lastly, a real-life application based on an interval-valued m-polar fuzzy competition graph has been discussed.

In a congested road network, it is always desirable to find a reliable shortest path so far as travel time is concerned. Physical distance and travel time are invariably uncertain due to large-scale maintenance work, natural disaster, heavy accident, weather conditions, etc. Many researchers considered travel time to be stochastic where the Bellman’s principle of optimality is not preserved. Also, any sub-path of a shortest path is not necessarily shortest. Some researchers find non-dominated paths under different variants as a multi-objective optimization problem. Moreover, in the absence of sample data, probability theory cannot be used. In this work, the travel time is considered to be uncertain variable, where no sample data is required and it is established that Bellman’s principle of optimality is preserved. So, in contrast to the stochastic environment, classical dynamic approach can be used to find the α-reliable shortest path using uncertainty theory.

The valency based topological indices (TIs) are defined by algebraic functions as the chemical graph theory tools in which the structural parameters are used as input and the output is related with the topology of chemical species. In theoretical chemistry, the TIs are mainly used to investigate/develop the QSAR and QSPR investigations of the molecular graphs. The Reformulated F-index or RFI is one such kind of TIs. The RFI for a molecular graph Y is defined as RFY=∑uv∈EYdYu+dYv−23, where dYu is the degree of a vertex u in Y. In this paper, we study the RFI for vertex and edge version of F-join of given general graphs which are related to subdivision and total graph.

In mathematical chemistry, the topological indices with highly correlation factor play a leading role specifically for developing crucial information in QSPR/QSAR analysis. Recently, there exists a new graph invariant, namely, Y-index of graph proposed by Alameri as the sum of the fourth power of each and every vertex degree of that graph. The approximate range of the descriptors is determined by obtaining the bounds for the topological indices of graphs. In this paper, firstly, some upper bounds for the Y-index on trees with several types of domination number are studied. Secondly, some new bounds are also presented for this index of graphs in terms of relevant parameters with other topological indices. Additionally, a new idea on bounds for the Y-index by applying binary graph operations is computed.

The study of the inverse problem (IP) based on the topological indices (TIs) deals with the numerical relations to TIs. Mathematically, the IP can be expressed as follows: given a graph parameter/TI that assigns a non-negative integer value g to every graph within a given family G of graphs, find some G∈G for which TIG=g. It was initiated by the Zefirov group in Moscow and later Gutman et al. proposed it. In this paper, we have established the IP only for the Y-index, Gourava indices, second hyper-Zagreb index, reformulated first Zagreb index, and reformulated F-index since they are closely related to each other. We have also studied the same which is true for the molecular, tree, unicyclic, and bicyclic graphs.

Fuzzy set is a modern tool for depicting uncertainty. This paper introduces the concept of fuzzy sub e-group as an extension of fuzzy subgroup. The concepts of identity and inverse are generalized in fuzzy sub e-groups. Every fuzzy subgroup is proven to be a fuzzy sub e-group, but the converse is not true. Various properties of fuzzy sub e-groups are established. Moreover, the concepts of proper fuzzy sub e-group and super fuzzy sub e-group are discussed. Further, the concepts of fuzzy e-coset and normal fuzzy sub e-group are presented. Finally, we describe the effect of e-group homomorphism on normal fuzzy sub e-groups.

Topological indices (TIs) are expressed by constant real numbers that reveal the structure of the graphs in QSAR/QSPR investigation. The reformulated second Zagreb index (RSZI) is such a novel TI having good correlations with various physical attributes, chemical reactivities, or biological activities/properties. The RSZI is defined as the sum of products of edge degrees of the adjacent edges, where the edge degree of an edge is taken to be the sum of vertex degrees of two end vertices of that edge with minus 2. In this study, the behaviour of RSZI under graph operations containing Cartesian product, join, composition, and corona product of two graphs has been established. We have also applied these results to compute RSZI for some important classes of molecular graphs and nanostructures.

In this study, the concept of picture fuzzy competition graph along with its two subclasses such as picture fuzzy k-competition graphs and p-competition picture fuzzy graphs are introduced. Picture fuzzy competition graph is one of the generalization of competition graph. Several properties of picture fuzzy competition graphs have been investigated. Some new types of picture fuzzy graphs have been introduced like picture fuzzy open neighborhood graph and picture fuzzy closed neighborhood graph. In addition, the relationship between picture fuzzy k-competition graph and picture fuzzy k-neighborhood graph are established. An application of picture fuzzy competition graph in medical science is presented.