Jurnal Diferensial

ABC Index Analysis: Physical Properties of Prenylated Xanthone

Lia Fitta Pratiwi — 2025-11-01

Xanthone is a heterocyclic compound with various substituents (hydroxy, prenyl, geranyl, methoxy, halogens, and others). The presence of these substituents contributes to diverse biological activities, including anticancer, antidiabetic, and antioxidant properties. This study aims to optimize the design of xanthone derivatives through a mathematical approach using Chemical Topological Graphs (CTG) and the Atom-Bond Connectivity (ABC) index. A literature review was conducted to identify the physicochemical properties, biological activities, and molecular structures of compounds such as 1,7-dihydroxy-3-methoxy-2-(3-methylbut-2-enyl)xanthone (1), Gartanin (2), and Garcinon (3). These xanthone derivatives are distinguished by the number of prenyl substitutions on their core structures. Chemical graph theory is employed to represent molecular structures, with atoms represented as nodes and chemical bonds as edges. The ABC index is calculated based on the degree of connected atoms within the molecules and correlated with the compounds’ physicochemical properties and bioactivity. The ABC index values for compounds (1), (2), and (3) are 32.186, 43.987, and 51.744, respectively. These values indicate that an increase in prenyl substitutions leads to higher ABC index values, which correspond to decreased polarity, increased boiling points, and enhanced bioactivity and stability of the xanthone derivatives

The Metric Dimension of Theta Graph

Deddy Rahmadi — 2025-11-01

Let $ G = (V, E) $ be a connected graph with vertex set $ V(G) $ and edge set $ E(G) $. For any two vertices $ u $ and $ v $ in $ G $, the shortest path distance between $ u $ and $v$ is denoted by $d(u, v)$. If $W = \{w_1, w_2, \dots, w_k\}$ is an ordered set of vertices in the connected graph $G$ , and $v \in V(G)$, then the representation of vertex $v$ with respect to $W$ , denoted as $r(v|W)$, is $r(v|W) = (d(v, w_1), d(v, w_2), \dots, d(v, w_k))$. If $r(v|W)$ is distinct for each vertex $v \in V(G)$, then $W$ is referred to as a resolving set for $G$. The resolving set with the smallest cardinality is called the minimum resolving set, and the cardinality of this set is the metric dimension of $G$, denoted by $\dim(G)$. This paper explores the metric dimension of the theta graphs.

Sombor Indices of Prime Coprime Graph for Integer Modulo Group

Abdurahim Abdurahim — 2025-11-01

The prime coprime graph of integers modulo n is an ordered pair consisting of a set of vertices (integers modulo n) and edges. Two distinct vertices are said to be adjacent if the greatest common divisor (gcd) of their orders is either 1 or a prime number. This article discusses the prime coprime graph of integers modulo n for n = pq, where p < q are prime numbers. The results of the study include the degree characteristics of the vertices and the subgraphs formed. Additionally, the Sombor index of the graph is also determined.

A Novel Approach to Topological Indices of the Identity Graph Associated with the Finite Group

Abdul Gazir Syarifudin — 2025-11-01

Graph theory is applied to study network structures in various disciplines, such as computer science and discrete mathematics. The combination of graphs and algebra has become a widely discussed topic in research within the fields of algebra and combinatorics. Research on group representations on graphs and topological indices has been extensively conducted, one such example is on the identity graph. A identity graph of a group $G$ which is an ordered pair $V(G)$ and $E(G)$, where all elements of $G$ serve as vertices, and two vertices $x,y \in G$ are adjacent if and only if $x*y=e$. This study proposes an alternative approach to calculating topological indices in the identity graph of the multiplicative group of integers modulo n.

Klasterisasi Produktivitas Daerah di Jawa Tengah Berdasarkan Ketenagakerjaan Menggunakan K-Means dan Average Linkage

Ahmad Firqi Nashrullah — 2025-11-01

This study employs K-Means and Agglomerative Clustering (Average Linkage) to group regions based on variables such as the number of residents, unemployment rate, and other supporting indicators. The data are normalized and evaluated using the Silhouette Score metric, yielding three optimal clusters. Average Linkage (0.3596) outperforms K-Means (0.2627). The Average Linkage results indicate that cluster 1 is characterized by stable productivity and low unemployment, cluster 2 consists solely of Semarang City with the highest Human Development Index and wages, and cluster 3 comprises underdeveloped areas with high unemployment and low wages. This clustering is highly beneficial for supporting more targeted data-driven regional development policies.

Implementasi Runtun Waktu Samar Orde Tinggi Yolcu-Egrioglu-Aladag dalam Melakukan Peramalan

Luthfia Amanah — 2025-11-01

Forecasting plays an important role in strategic decision-making in various fields, ranging from economics to public management. However, traditional forecasting methods such as ARIMA and linear regression have limitations in handling the characteristics of time series data that are often non-linearly patterned, non-stationary, and experience sudden fluctuations. This research proposes a fuzzy time series (FTS) approach that is more flexible in modeling uncertainty and complex patterns without relying on the assumption of linearity. The purpose of this research is to forecast the population in Sleman Regency using the Yolcu-Egrioglu-Aladag (YEA) higher order fuzzy time series. The data used is annual data on the population of Sleman Regency from 1998 to 2024, a total of 27 observational data divided into 80% training data and 20% testing data. The YEA FTS uses a slice operation to simplify inputs in higher-order models, fuzzification with Fuzzy C-Means and forming fuzzy relations through a Feedforward Neural Network with the Levenberg-Marquardt algorithm, and defuzzification. The results show that the YEA model produces a small Mean Absolute Percentage Error (MAPE) value on the testing data, so it is good for forecasting the population in Sleman Regency. The MAPE value of the YEA FTS is 2.22% (training) and 1.00% (testing).

Keywords: : Forecasting, Fuzzy Time Series, High Order Yolcu-Egrioglu-Aladag, Fuzzy C-Means, Feedforward Neural Network, Population

Analisis Hubungan Nilai Indeks Zagreb dengan Titik Didih dan Kestabilan Senyawa Kerangka Dasar Calamenene

M. Ibnu Sabil — 2025-11-01

A new compound, dysoxyphenol, isolated from Dysoxylum densiflorum, has a
calamenene-based framework. The structure of dysoxyphenol is classified as a sesquiterpene. This
study discusses the relationship between Zagreb Index values and the physical properties and
stability of two compounds: calamenene and dysoxyphenol. The Zagreb Index, consisting of the
first and second indices, is used to measure the structural stability of a graph through the sum of
squared vertex degrees. The objective of this analysis is to determine the Zagreb Index values for
calamenene and dysoxyphenol, to measure the boiling points and stability of the compounds. This
analysis was carried out by examining the structures of calamenene and dysoxyphenol. The obtained
structures were then analyzed graphically. The results of this study show that the first and second
Zagreb Index values for calamenene are 262 and 340, which are smaller compared to the first and
second Zagreb Index values for dysoxyphenol, which are 264 and 350, respectively. This indicates
that calamenene has a lower boiling point than dysoxyphenol and is more stable than dysoxyphenol.

Evaluasi Kinerja Uji Normalitas pada Ragam Distribusi dan Ukuran Sampel

Shindi Shella May Wara — 2025-11-01

The normal distribution is a fundamental assumption in many parametric statistical methods. Therefore, testing for data normality is a crucial step prior to further analysis. This study aims to evaluate the performance of three widely used normality test methods: Kolmogorov-Smirnov (KS), Anderson-Darling (AD), and Shapiro-Wilk (SW), across various distributions (standard normal, exponential, and t-student with degrees of freedom 1, 20, and 100) and sample sizes (n = 20, 50, 100, 200, and 500). Data were generated through simulation with 1000 iterations for each combination. The results show that the KS method performs well on standard normal and t-student distributions with larger degrees of freedom. The AD method proves to be more sensitive, especially in detecting deviations from normality, though it is less stable for small sample sizes. Meanwhile, the SW method demonstrates optimal performance with large samples. These findings provide practical guidance in selecting appropriate normality test methods based on the characteristics of the data.

Nonlocal Metric Dimension of Windmill Graph

Fithri Annisatun Lathifah — 2025-11-01

Let G = (V (G), E(G)) be a simple and connected graph. The distance between two vertices u and v
in G, is the length of a shortest path from u to v, denoted by d(u, v). Suppose S = {s1, s2, ...sk} is an
ordered subset of vertices of G, then the metric representation of a vertex u ∈ V (G) with respect to S,
denoted by r(u|S), is the k−vector (d(u, s1), d(u, s2), ..., d(u, sk)). If every two nonadjacent vertices of
G have distinct metric representations with respect to S, then the set S is called a nonlocal resolving
set for G. A nonlocal resolving set with minimum cardinality is called a nonlocal metric basis. The
nonlocal metric dimension of G is the cardinality of the nonlocal metric basis of G and is denoted by
nldim(G). In this paper, we obtained nonlocal metric dimension of windmill graph.

Modelling Immunological Effects on Fractional Order of Cholera Dynamics with Behavioral Response via Numerical simulation

Mutairu Kayode Kolawole — 2025-11-01

Cholera, spread by the bacterium Vibrio cholerae, is still a major health problem in places with unsanitary conditions. The way it spreads relies on the host’s immunity, certain environmental aspects and how clean people keep themselves and their properties. The model in this study applies Caputo fractional-order derivatives to capture the immunity of people, their hygiene, memory in diseases and various ways of controlling them. It includes the study of how people respond and interact with their environment and disease-related factors in a mathematical way. We perform solid analyses on the model, confirming the existence, uniqueness, positivity and boundedness of its solutions. A basic reproduction number is calculated to find out if the disease will continue to exist in a population. Analyzing what makes a disease-free state or an endemic equilibrium stable tells us how to best control the disease. Using the Laplace-Adomian Decomposition Method for solving the nonlinear fractional system results in simulations that match actual cholera behavior. Findings point out that a decline in immunity and better hygiene help reduce how cholera spreads. The framework supports an understanding of cholera spread and is also useful for examining other diseases that are highly complex.