The paper reviews the applications of eigen value in different areas. One of the area is in the analysis of graphs coming from networks. The development of theory regarding the eigenvalues and its maximum eigenvalue of the adjacency matrix arising from a general graph is already well-established. Here we review some notions from different context, i.e. led by observation of some simple experiments regarding the relation between graph, cliques, and the eigen values of the adjacency matrix. We focus on regular graphs having one or more cliques in their graph structures. We do some numerical experiment on the computation of the eigen values of the adjacency matrix and show some patterns on the relation between the structure of the graph (e.g. the maximum cliques, chromatic number) and the eigen values of the adjacency matrix. By observing these patterns we find some conclusion, such as: i. the maximum clique of a complete graph is given by the largest eigen value plus one; ii. the maximum clique of a cycle graph (simple incomplete regular graph) equals the largest eigen value, in which the value is two; iii. the maximum multiplicity of the eigen values of a cycle graph is two. Future direction of the development is also presented based on the careful analysis of the existing development.

Investment decision-making is often characterized by uncertainty and the subjective weighting of criteria. This study aims to develop a more robust decision support framework by integrating the Generalized Interval-Valued Hesitant Intuitionistic Fuzzy Soft Set (GIVHIFSS) with the Analytic Hierarchy Process (AHP) to objectively weight criteria and handle multi-evaluator hesitancy. In the proposed GIVHIFSS-AHP model, the AHP is employed to derive mathematically consistent criterion weights, which are subsequently embedded into the GIVHIFSS structure to accommodate interval-valued and hesitant evaluations from multiple decision-makers. The model is applied to a numerical case study evaluating five investment alternatives. Its performance is assessed through a comparative analysis with standard GIVHIFSS and GIFSS models, as well as a sensitivity analysis. The results indicate that the model produces financially rational rankings, identifying blue-chip technology stocks as the optimal choice (score: +2.4). The comparative analysis confirms its superiority over existing models, which yielded less-stable rankings. Moreover, the sensitivity analysis demonstrates the robustness of the results against minor perturbations in criterion weights. This research introduces a novel and synergistic integration of the AHP and GIVHIFSS. The key advantage of this approach lies in its ability to address the long-standing issue of arbitrary criterion weighting in Fuzzy Soft Set models by embedding the AHP as a foundational mechanism for ensuring validation and objectivity. This integration results in mathematically derived, consistent weights, thereby yielding empirically validated, more reliable, and defensible decision outcomes compared with existing models.

The Citarum River Basin (DAS Citarum) in Indonesia faces significant challenges in waste management, necessitating a circular economy-based approach to reduce land-based pollution, which is critical for achieving the sustainability goals of the blue economy in the basin. This study addresses the complexity and inherent uncertainty in decision-making processes related to this challenge by developing a novel hybrid model, namely the Weighted Fuzzy N-Soft Set combined with the COmbinative Distance-based Assessment (CODAS) method. The model synergistically integrates the weighted 10R strategies in the circular economy, obtained via the Analytical Hierarchy Process (AHP), the capability of Fuzzy N-Soft Sets to represent uncertainty granularly, and the robust ranking mechanism of CODAS. Applied to a case study covering 16 types of waste in the Citarum River Basin, the model effectively processes expert assessments that are ambiguous regarding the 10R criteria. The results indicate that single-use plastics, particularly plastic bags (HDPE), styrofoam, transparent plastic sheets (PP), and plastic cups (PP), are the top priorities for intervention, in line with the high AHP weights for upstream strategies such as Refuse (0.2664) and Rethink (0.2361). Comparative analysis with alternative models, namely Fuzzy N-Soft Set-CODAS, Weighted Fuzzy N-Soft Set with row-column sum ranking, and Weighted Fuzzy N-Soft Set-TOPSIS, confirms the superiority of the proposed hybrid model in producing ecologically rational priorities, free from purely economic value biases. Further sensitivity analysis shows that the model remains highly robust across various weighting scenarios. This study concludes that the WFN-SS-CODAS framework provides a rigorous, data-driven, and reliable decision support tool for translating circular economy principles into actionable waste management priorities, directly supporting the restoration and sustainability goals of the blue economy in river basins. The findings suggest that targeting the high-priority waste types identified by the model addresses the dominant fraction of riverine pollution, indicating the potential for significant waste volume reduction. This research was conducted to directly contribute to achieving multiple targets under SDG 6 (Clean Water and Sanitation), SDG 12 (Responsible Consumption and Production), and SDG 14 (Life Below Water).

The main objective of this study is to determine the existence and uniqueness of solutions to the fractional Black–Scholes equation. The solution to the fractional Black–Scholes equation is expressed as an infinite series of converging Mittag-Leffler functions. The method used to discover the new solution to the fractional Black–Scholes equation was the Daftardar-Geiji method. Additionally, the Picard–Lindelöf theorem was utilized for the existence and uniqueness of its solution. The fractional derivative employed was the Caputo operator. The search for a solution to the fractional Black–Scholes equation was essential due to the Black–Scholes equation’s assumptions, which imposed relatively tight constraints. These included assumptions of a perfect market, a constant value of the risk-free interest rate and volatility, the absence of dividends, and a normal log distribution of stock price dynamics. However, these assumptions did not accurately reflect market realities. Therefore, it was necessary to formulate a model, particularly regarding the fractional Black–Scholes equation, which represented more market realities. The results obtained in this paper guaranteed the existence and uniqueness of solutions to the fractional Black–Scholes equation, approximate solutions to the fractional Black–Scholes equation, and very small solution errors when compared to the Black–Scholes equation. The novelty of this article is the use of the Daftardar-Geiji method to solve the fractional Black–Scholes equation, guaranteeing the existence and uniqueness of the solution to the fractional Black–Scholes equation, which has not been discussed by other researchers. So, based on this novelty, the Daftardar-Geiji method is a simple and effective method for solving the fractional Black–Scholes equation. This article presents some examples to demonstrate the application of the Daftardar-Gejji method in solving specific problems.

Twitter is a social network that is very popular among the people that allows the users to send and read text-based messages up to 140 characters, known as tweets. Since it was first launched until now, this social network has become one of the ten most visited sites on the internet. Because there are so many people use this social network, Twitter has become a source of information that can disseminate information in a short time. Dissemination of information on Twitter is very preferred by the accounts of Twitter users. The most important account in the dissemination of information on Twitter can be searched by calculating the centrality measurement. In this paper, four centrality measures are used to find the most important accounts, that are degree centrality, closeness centrality, betweenness centrality, and eigenvector centrality. Before calculating the centrality of the Twitter accounts, data must be collected first, wherein this paper the data to be used is tweeted data from Twitter using hashtag #BersatuLawanCorona obtained using Python with the help of Twitter API. In calculating this centrality value, we can get the most influential account in distributing information related to hashtag #BersatuLawanCorona. Based on the data, obtained one account that became the most influential account because it obtained the highest value for each centrality measurement, which is SapawargaSby account.

The Millennium Development Goals (MDGs), which began in 2000 with 8 goal points, have not been able to solve the global problems. The MDGs were developed into Sustainable Development Goals (SDGs) in 2015 with 17 targeted goal points achieved in 2030. Until now, methods for determining the priority of SDGs are still attractive to researchers. Centrality measure is one of the tools in determining the priority goal points on a network by using graph theory. There are four measurements of centrality used in this paper, namely degree centrality, betweenness centrality, closeness centrality, and eigenvector centrality. The calculation results obtained from the four measurements are compared dan analyzed, to conclude which goal points are the most prior and the least prior. Furthemore, in this paper we present other example with simple graph to show that each different centrality calculation possibly resulted different priority node, the calculation of this illustration is done using a Python’s library named NetworkX