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The directed rough fuzzy graph (DRFG) is a fusion of rough and fuzzy theory, as it deals with incomplete and vague information simultaneously. Connection or the strength of connectivity (SC)is vital in the realm of circuits or networks that are linked to the real world. As a result, SCis one of the most essential aspects of a directed rough fuzzy network system. The neighborhood connectivity index (NCI)is one such parameter that has a variety of applications in network theory. In this study, our main objective is to present a new topological index NCIbased on DRFGs to solve complicated problems. Motivated by the modeling of networks, the strength of vertices to their neighboring vertices, and the efficiency of DRFGs to solve complex problems, we aim to study the NCI of DRFGs. In this paper, we successfully introduce a notion NCI based on DRFGs to deal with the uncertainties that arise in real-world problems. Based on the strength of vertices to neighboring vertices, we provide several lower and upper bounds for the NCIof DRFGs with reference to other graph invariants such as the number of vertices, edges, and degree distance. When we study NCIin operations for DRFGs with a large number of vertices, the degree of vertices in a DRFG provides a confusing picture. Therefore, a mechanism to determine the NCIfor DRFG operations is therefore required. Therefore, generalized formulas for the NCIof DRFGs obtained by operations such as union, composition, and Cartesian product are also developed. An algorithm for obtaining the NCIof DRFGs is also proposed. Further, an application of the NCIof DRFGs in traffic flow networks was discussed to identify the busiest intersection using the proposed algorithm. Finally, we illustrate a comparative analysis and analysis table of the established approach (NCI) with existing techniques (connectivity index (CI)and Wiener index (WI)) are shown to demonstrate the validity of the presented approach.

Fuzzy soft graphs are effective mathematical tools that are used to model the vagueness of the real world. A fuzzy soft graph is a fusion of the fuzzy soft set and the graph model and is widely used across different fields. In this current research, the concept of picture fuzzy soft graphs is presented by combining the theory of picture soft sets with graphs. The introduction of this new picture fuzzy soft graphs is an emerging concept that can be rather developed into various graph theoretical concepts. Since soft sets are most usable in real-life applications, the newly combined concepts of the picture and fuzzy soft sets will lead to many possible applications in the fuzzy set theoretical area by adding extra fuzziness in analyzing. The notions of picture soft graphs, strong and complete picture soft fuzzy graphs, a few types of product picture fuzzy soft graphs, and regular, totally regular picture fuzzy soft graphs are discussed and validated using real-world scenarios. In addition, an application of decision-making for medical diagnosis in the current COVID scenario using the picture fuzzy soft graph has been illustrated.

Over the past few years, we have seen the spread of wireless networks in institutions, businesses, and private homes. Their wide variety, market availability, and ease of implementation have made this wireless technology more popular than wired networks. Nevertheless, enterprises are still demanding better quality of service to meet the needs of sensitive applications and avoid data loss. The SDN approach provides the ability to manage QoS dynamically via a controller. In this paper, we developed a secure model based on dynamic access control and the detection of unreliable access points. We also proposed QoS management in a wireless network by adopting the SDN approach to address connection problems and efficiently utilize resources. By employing these tools and techniques, we can narrow down all possible options to one single best choice. By comparing the results, our model validates its effectiveness in minimizing latency, jitter, packet loss, and transaction delay and offers a good opinion score.

An elongation of the novel intuitionistic fuzzy set is a picture fuzzy set theory. The demonstration of this has been used to deal with the abstinence criteria in a decision-making problem. The uncertainty in nature occurs sometimes in real-world problems and amidst them, the prominent one is the shortest path problem (SPP) solving. In the last few years, one of the best algorithms on the network for finding SPP is Bellman–Ford. Due to uncertainty in the decision-making process, it becomes difficult for decision-makers for communicating their point of view and judgment with certainty. We conceive of SPP in this contribution via Bellman's algorithm (BA) for a network with trapezoidal picture fuzzy numbers (TPFNs). We introduce a new algorithm to stand the shortest picture fuzzy path between each pair of nodes. A TPFN is considered for the length of all edges. A numerical example for the validation of the presented algorithm has also been proposed. There has also been relative research with existing techniques showing the benefits of the new algorithm.

In the last decade, concealed by uncertain atmosphere, many algorithms have been studied deeply to workout the shortest path problem. In this paper, we compared the shortest path problem with various existing algorithms. Finally, we concluded the best algorithm for certain environment.

An elongation of the single-valued neutrosophic set is an interval-valued neutrosophic set. It has been demonstrated to deal indeterminacy in a decision-making problem. Real-world problems have some kind of uncertainty in nature and among them; one of the influential problems is solving the shortest path problem (SPP) in interconnections. In this contribution, we consider SPP through Bellman’s algorithm for a network using interval-valued neutrosophic numbers (IVNNs). We proposed a novel algorithm to obtain the neutrosophic shortest path between each pair of nodes. Length of all the edges is accredited an IVNN. Moreover, for the validation of the proposed algorithm, a numerical example has been offered. Also, a comparative analysis has been done with the existing methods which exhibit the advantages of the new algorithm.

Real-life decision-making problem has been demonstrated to cover the indeterminacy through single valued neutrosophic set. It is the extension of interval valued neutrosophic set. Most of the problems of real life involve some sort of uncertainty in it among which, one of the famous problem is finding a shortest path of the network. In this paper, a new score function is proposed for interval valued neutrosophic numbers and SPP is solved using interval valued neutrosophic numbers. Additionally, novel algorithms are proposed to find the neutrosophic shortest path by considering interval valued neutrosophic number, trapezoidal and triangular interval valued neutrosophic numbers for the length of the path in a network with illustrative example. Further, comparative analysis has been done for the proposed algorithm with the existing method with the shortcoming and advantage of the proposed method and it shows the effectiveness of the proposed algorithm.

AbstractIn this paper, inspired by the concept of generalized single-valued neutrosophic graphs(GSVNG) of the first type, we define yet another generalization of neutrosophic graph called the generalized interval-valued neutrosophic graph of 1 type (GIVNG1) in addition to our previous work on complex neutrosophic graph (CNG1) in [47]. We will also show a matrix representation for this new generalization. Many of the fundamental properties and characteristics of this new concept is also studied. Like the concept CNG1 in [47], the concept of GIVNG1 is another extension of generalized fuzzy graphs 1 (GFG1) and GSVNG1.

In this paper, we discuss the minimum spanning tree (MST) problem of an undirected neutrosophic weighted connected graph in which a single-valued neutrosophic number, instead of a real number/fuzzy number, is assigned to each arc as its arc length. We define this type of MST as neutrosophic minimum spanning tree (NMST). We describe the utility of neutrosophic numbers as arc lengths and its application in different real world MST problems. Here, a new algorithm for designing the MST of a neutrosophic graph is introduced. In the proposed algorithm, we incorporate the uncertainty in Kruskal algorithm for designing MST using neutrosophic number as arc length. A score function is used to compare different NMSTs whose weights are computed using the addition operation of neutrosophic numbers. We compare this weight of the NMST with that of an equivalent classical MST with real numbers as arc lengths. Compared with the existing algorithms for NMST, the proposed algorithm is more efficient due to the fact that the addition operation and the ranking of neutrosophic number can be done in straightforward manners. The proposed algorithm is illustrated by numerical examples.

This main purpose of this paper is to develop an algorithm to find the shortest path on a network in which the weights of the edges are represented by bipolar neutrosophic numbers. Finally, a numerical example has been provided for illustrating the proposed approach.

In this paper, we first define the concept of bipolar single neutrosophic graphs as the generalization of bipolar fuzzy graphs, N-graphs, intuitionistic fuzzy graph, single valued neutrosophic graphs and bipolar intuitionistic fuzzy graphs.

Controlling traffic flow on roads is an important traffic management task necessary to ensure a peaceful and safe environment for people. The number of cars on roads at any given time is always unknown. Type-2 fuzzy sets and neutrosophic sets play a vital role in dealing efficiently with such uncertainty. In this paper, a triangular interval type-2 Schweizer and Sklar weighted arithmetic (TIT2SSWA) operator and a triangular interval type-2 Schweizer and Sklar weighted geometric (TIT2SSWG) operator based on Schweizer and Sklar triangular norms have been studied, and the validity of these operators has been checked using a numerical example and extended to an interval neutrosophic environment by proposing interval neutrosophic Schweizer and Sklar weighted arithmetic (INSSWA) and interval neutrosophic Schweizer and Sklar weighted geometric (INSSWG) operators. Furthermore, their properties have been examined; some of the more important properties are examined in detail. Moreover, we proposed an improved score function for interval neutrosophic numbers (INNs) to control traffic flow that has been analyzed by identifying the junction that has more vehicles. This improved score function uses score values of triangular interval type-2 fuzzy numbers (TIT2FNs) and interval neutrosophic numbers.