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Social networks are absolutely a useful and important place for connecting people within the world. A basic issue in a social network is to identify the key persons within it. This is why different centrality measures have been found over the years. In this survey paper, we present past and present research works on measures of centrality in social network. For this plan, we discuss mathematical definitions and different developed centrality measures. We also present some applications of centrality measures in biology, research, security, traffic, transportation, drug, class room. At last, our future research work on centrality measure is given.

SARS-CoV-2 is an RNA virus that was identified for the first time in December 2019 in Wuhan, China. The World Health Organization (WHO) labeled the novel coronavirus (COVID-19) outbreak a worldwide pandemic on March 11, 2020, due to its widespread infectivity pattern. Because of the catastrophic COVID-19 outbreak, the development of safe and efficient vaccinations has become a key priority in every health sector throughout the globe. On the 13th of January 2021, the vaccination campaign against SARS-CoV-2 was launched in India and started the administration of two types of vaccines known as Covaxin and Covishield. Covishield is an adenovirus vector-based vaccine, and Covaxin was developed by a traditional method of vaccine formulation, which is composed of adjuvanted inactivated viral particles. Each vaccine’s utility or efficiency is determined by its formulation, adjuvants, and mode of action. The efficacy of the vaccination depends on numeral properties like generation antibodies, memory cells, and cell-mediated immunity. According to the third-phase experiment, Covishield showed effectiveness of nearly 90%, whereas Covaxin has an effectiveness of about 80%. Both vaccination formulations in India have so far demonstrated satisfactory efficacy against numerous mutant variants of SARS-CoV-2. The efficacy of Covishield may be diminished if the structure of spike (S) protein changes dramatically in the future. In this situation, Covaxin might be still effective for such variants owing to its ability to produce multiple antibodies against various epitopes. This study reviews the comparative immunogenic and therapeutic efficacy of Covaxin and Covishield and also discussed the probable vaccination challenges in upcoming days.

The computation of link prediction is one of the most important tasks on a social network. Several methods are available in the literature to predict links in networks and RSM index is one of them. The RSM index is applicable in the fuzzy environment and it does not incorporate the notion of falsity and indecency parameters which occur frequently in uncertain environments. In the present method, the behaviors of the common neighbor and the other parameters, like nature of job, location, etc., are considered. In this paper, more parameters are included in the RSM index for making it more flexible and realistic and it is best fitted in the neutrosophic environment. Many important properties are studied for this modified RSM index. A small network from Facebook is considered to illustrate the problem.

Graph colouring enjoys many practical as well as theoretical uses. Graph colouring is still a very active subject of research. This article introduces a new concept of the chromatic number of the neutrosophic graph (NG). There is a discussion of the difference between the fuzzy colouring of the fuzzy graph (FG) and the NG. Finally, a real-life application for detecting phishing websites has been demonstrated by NG colouring.

To represent qualitative aspect of uncertainty and imprecise information, linguistic preference relation (LPR) is a powerful tool for experts expressing their opinions in group decision-making (GDM) according to linguistic variables (LVs). Since for an LV, it generally means that membership degree is one, and non-membership and hesitation degrees of the experts cannot be expressed. Pythagorean linguistic numbers/values (PLNs/PLVs) are novel choice to address this issue. The aim of this paper which we propose a GDM problem involved a large number of the experts is called large-scale GDM (LSGDM) based on Pythagorean linguistic preference relation (PLPR) with a consensus model. Sometimes, the experts do not modify their opinions to achieve consensus. Therefore, the experts’ proper opinions’ management with their non-cooperative behaviors (NCBs) is necessary to establish a consensus model. At the same time, it is essential to ensure the proper adjustment of the credibility information. The proposed model using grey clustering method is divided with the experts’ similar evaluations into a subgroup. Then, we aggregate the experts’ evaluations in each cluster. A cluster consensus index (CCI) and a group consensus index (GCI) are presented to measure consensus level among the clusters. Then, we provide a mechanism for managing the NCBs of the clusters, which contain two parts: (1) NCB degree is defined using CCI and GCI for identifying the NCBs of the clusters; (2) implemented the weight punishment mechanism of the NCBs clusters to consensus improvement. Finally, an example is offered for usefulness of the proposed approach.

Secure communication is essential in today’s rapidly evolving digital environment, and strong encryption methods are required to protect private data from unwanted access. The aim of this study is to strengthen the security and complexity of encrypted communications by adopting a new form of cryptographic encryption technique based on the principles of an intuitionistic fuzzy graph. Key graph-theoretic measures, such as domination number, vertex categorization (alpha-strong, beta-strong, and gamma-strong), vertex order coloring, and chromatic number, play important roles in this process. Domination number finds the key vertices of the network, while vertex strength categorization and fuzzy graph coloring provide multiple encryption layers, hence the encoded message is highly resistant to decryption unless a proper key is used. The chromatic number offers further security through various patterns of vertex coloring. The comparative analysis shows the proposed approach to be superior compared to RSA, AES, ECC, and Blowfish due to its increased security, computational efficiency, and resilience to attacks. This framework can be applied to the protection of banking PINs, military access codes, government identification numbers, cryptographic keys, and medical records, so it is an extremely versatile solution for protecting sensitive data. This multi-step approach to encryption through the proposed technique ensures safe transfer and efficient encoding as it establishes a complicated framework.

Topological indices are the numbers that remain constant under graph automorphism. Topological indices describe a network’s connectivity, structure, and topological characteristics. These indices have many applications in crisp graphs. However, in many cases, it is observed that some situations can’t be described using the idea of crisp graphs. So, to overcome this issue, the need to define topological indices for fuzzy and bipolar fuzzy graphs arises. The F-index, or the Forgotten Index, is a significant topological index. A bipolar fuzzy graph with two opposite-sided opinions of both the edges and vertices measures the impreciseness or uncertainties of the edges and vertices along the positive and negative sides. In this article, we have presented the Forgotten Index for bipolar fuzzy graphs. Then, we have proved some theorems regarding the F-index of numerous types of bipolar fuzzy graphs, such as regular bipolar fuzzy graphs, complete bipolar fuzzy graphs, etc., the bounds of the F-index in bipolar fuzzy graphs, and the relationships of the F-index with other topological indices in bipolar fuzzy graphs. We have applied the proposed topological index, the F-index for bipolar fuzzy graphs, to matrimonial websites to find potential life partners based on compatibility and discussed the application of the Forgotten Index in gene regulatory networks.

This paper presents a comprehensive study of the linguistic Z-graph, which is a novel framework designed to analyze linguistic structures within social networks. By integrating concepts from graph theory and linguistics, the linguistic Z-graph provides a detailed understanding of language dynamics in online communities. This study highlights the practical applications of linguistic Z-graphs in identifying central nodes within social networks, which are crucial for online businesses in market capture and information dissemination. Traditional methods for identifying central nodes rely on direct connections, but social network connections often exhibit uncertainty. This paper focuses on using fuzzy theory, particularly linguistic Z-graphs, to address this uncertainty, offering more detailed insights compared to fuzzy graphs. Our study introduces a new centrality measure using linguistic Z-graphs, enhancing our understanding of social network structures.

To solve fractional differential equations, they are typically converted into their corresponding crisp problems through a process known as the embedding method. This paper introduces a novel direct approach to solving fuzzy differential equations using fuzzy calculations, bypassing the need for this transformation. In this study, we develop the fuzzy Adams–Bashforth (A-B) method and the fuzzy Adams–Moulton (A-M) method to find numerical solutions for fuzzy fractional differential equations (FFDEs) with fuzzy initial values. To demonstrate the accuracy and efficiency of the proposed methods, we determine both the local truncation error and the global truncation error. Additionally, we establish the convergence and stability of these methods in detail. Finally, numerical examples are provided to illustrate the flexibility and effectiveness of the proposed methods.

In the middle of November 2021, Omicron (B.1.1.529), a novel variant of SARS-CoV-2 was identified in South Africa. Owing to continuous increasing cases with rapid transmissibility and immune evasion, the World Health Organization (WHO) has categorized this strain as a variant of concern (VOC). In total, over 60 mutations have been identified in Omicron (BA.1) and latterly, its three sub-lineages (BA.1.1, BA.2, and BA.3) have also been found with additional mutations and pathogenicity. The highly contagious Omicron causes less severe sickness than Delta, but it is still dangerous for those who have not been vaccinated. Following the unique identification of the Omicron variant, a fresh debate has erupted regarding the natural vaccines. A number of experts believe that Omicron can work as a natural vaccine, because it is similar to live attenuated vaccines in certain ways. Additionally, it was highlighted that the high rate of antibody generation in individuals cured of Omicron provide suggestive evidence in favor of those researchers who claimed Omicron acts as natural vaccine. Some disagreements also noted, as it also has tremendous health effects and high infection rate, as similar to the prior variants. This review summarizes the contradictory scenario among the scientists about Omicron variant.