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All Journal JURNAL MATEMATIKA STATISTIKA DAN KOMPUTASI CAUCHY: Jurnal Matematika Murni dan Aplikasi Journal of the Indonesian Mathematical Society Jurnal Matematika & Sains JMPM: Jurnal Matematika dan Pendidikan Matematika BAREKENG: Jurnal Ilmu Matematika dan Terapan Unisda Journal of Mathematics and Computer Science (UJMC) JTAM (Jurnal Teori dan Aplikasi Matematika) Beta: Jurnal Tadris Matematika EIGEN MATHEMATICS JOURNAL Jurnal Abdi Insani Mandalika Mathematics and Educations Journal Contemporary Mathematics and Applications (ConMathA) InPrime: Indonesian Journal Of Pure And Applied Mathematics JPIn : Jurnal Pendidik Indonesia Majalah Ilmiah Matematika dan Statistika (MIMS) Jurnal Ilmiah Abdi Mas TPB Unram Jurnal Diferensial Griya Journal of Mathematics Education and Application Evolusi: Journal of Mathematics and Natural Sciences Prosiding Konferensi Nasional PKM-CSR MATHunesa: Jurnal Ilmiah Matematika Journal of Fundamental Mathematics and Applications (JFMA) Jurnal Pepadu Basis : Jurnal Ilmiah Matematika Jurnal Pengabdian Inovasi Masyarakat Indonesia Pemberdayaan Masyarakat: Jurnal Aksi Sosial Sinergi dan Harmoni Masyarakat MIPA
I Gede Adhitya Wisnu Wardhana
Department Of Mathematics, Faculty Of Mathematics And Natural Science, University Of Mataram
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Indeks Harmonik, Hyper-Wiener, Dan Randic Dari Graf Pangkat Pada Grup Dihedral Syafitri, Hanna; Wardhana, I Gede Adhitya Wisnu; Abdurahim, Abdurahim; Alfian, Muhammad Rijal; Syechah, Bulqis Nebulla
Basis : Jurnal Ilmiah Matematika Vol. 4 No. 2 (2025): BASIS: Jurnal Ilmiah Matematika
Publisher : Universitas Mulawarman

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.30872/xcf56e20

Abstract

Grup  disebut Grup Dihedral dengan order 2n, dinotasikan D2n, untuk bilangan asli n yang lebih besar dari 3 merupakan grup yang dibangun oleh dua element berbeda yaitu a dan b. Graf Pangkat dari grup dihedral D2n adalah graf yang semua simpulnya merupakan semua element dari D2n. Lebih jah, dua simpul berbeda, a dan b, dikatakan bertetangga jika dan hanya jika  ax = b atau by= a  dengan x dan y bilangan bulat positif. Pada artikel ini akan dikaji beberapa indeks topologi dari graf pangkat untuk grup dihedral, yaitu indeks Randic, Harmonik, dan Hyper-Wiener. Hasil penelitian ini berupa rumus umum dari indeks topologi dari graf pangkat grup dihedral.
CERTAIN INDEXES OF UNIT GRAPH IN INTEGER MODULO RINGS WITH SPECIFIC ORDERS Lestari, Sahin Two; Albaracin, Jimboy R.; Wisnu Wardhana, I Gede Adhitya
BAREKENG: Jurnal Ilmu Matematika dan Terapan Vol 19 No 4 (2025): BAREKENG: Journal of Mathematics and Its Application
Publisher : PATTIMURA UNIVERSITY

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.30598/barekengvol19iss4pp2455-2466

Abstract

Topological indices quantify structural properties of graphs and find wide applications in chemistry, physics, and network analysis. This study investigates several key indices—namely the Harary Index, Wiener Index, Randić Index, Schultz Index, and the Zagreb Indices—within the context of unit graphs derived from the ring of integers modulo. General formulas for these indices are established, demonstrating how they reflect the combinatorial and algebraic characteristics of unit graphs. Each index captures distinct structural aspects: the Wiener Index evaluates global connectivity and correlates with molecular stability and boiling points; the Randić Index highlights molecular branching relevant to enzyme activity; the Harary Index models electronic interactions through distance reciprocals; and the Zagreb Indices and Schultz Index provide insights into bonding properties and molecular interactions. By linking these indices to unit graphs, this work reinforces the synergy between graph theory and algebra, offering a systematic framework to interpret algebraic structures through graph-based invariants. The results not only contribute to theoretical understanding but also suggest potential applications in modeling chemical compounds and complex networks, paving the way for further exploration of topological indices in other algebraically defined graphs.
ENERGY OF NON-COPRIME GRAPH ON MODULO GROUP Karang, Gusti Yogananda; Wisnu Wardhana, I Gede Adhitya; Angamuthu, Manimaran
BAREKENG: Jurnal Ilmu Matematika dan Terapan Vol 19 No 4 (2025): BAREKENG: Journal of Mathematics and Its Application
Publisher : PATTIMURA UNIVERSITY

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.30598/barekengvol19iss4pp2937-2952

Abstract

A graph is a mathematical structure consisting of a non-empty set of vertices and a set of edges connecting these vertices. In recent years, extensive research on graphs has been conducted, with one of the intriguing topics being the representation of graphs within algebraic structures, particularly groups. This approach bridges two areas of mathematics: graph theory and algebra. This study focuses on graph representation, specifically non-coprime graphs in the group of integers modulo ​, where , is a prime number, and is a non-negative integer. The non-coprime graph of a group is defined as a graph with the vertex set , where is the identity element of . Two distinct vertices and are connected by an edge if . Specifically, this research investigates the Sombor energy, the Degree Sum energy, the Degree Exponent Sum energy, the Laplacian energy, the Distance Laplacian energy, and the Distance Signless Laplacian energy of a non-coprime graph on a modulo group.
The First Zagreb Index and the Narumi-Katayama Index of the Non-Commuting Graph on the Group U_{6n} Hisan, Khairatun; Gambo, Ibrahim; Wardhana, I Gede Adhitya Wisnu
InPrime: Indonesian Journal of Pure and Applied Mathematics Vol. 7 No. 2 (2025)
Publisher : Department of Mathematics, Faculty of Sciences and Technology, UIN Syarif Hidayatullah

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.15408/sxha8875

Abstract

This paper introduces a novel approach to computing the First Zagreb and Narumi–Katayama indices for non-commuting graphs associated with specific algebraic groups, specifically focusing on the group U_{6n}. The Narumi–Katayama index, first introduced by Narumi and Katayama in 1984, is a degree-based topological index widely used in the study of various graph properties, including its applications in theoretical chemistry. Non-commuting graphs, where two elements are adjacent if and only if they do not commute, have become an intriguing object of study in recent years. To the best of our knowledge, this is the first study to derive closed-form expressions for the First Zagreb and Narumi–Katayama indices on the non-commuting graph of the group U_{6n}. Building on previous research on the detour index and eccentric connectivity in the graph Γ(U_{6n}), this work makes new contributions by deriving generalized formulas that apply to a broader class of non-commutative groups. Unlike previous studies that focused on commuting or coprime graphs, this research specifically addresses the structure and index computation of non-commuting graphs in a group-theoretic context. The findings offer new theoretical insights into algebraic graph theory by linking degree-based indices with the internal structure of non-abelian groups. These results are expected to expand the understanding of the topological properties of non-commuting graphs and to provide valuable connections to chemical applications.Keywords: Non-commuting graph; First Zagreb Index; Narumi–Katayama Index; graph topology; group structure. AbstrakArtikel ini memperkenalkan pendekatan baru untuk menghitung indeks First Zagreb dan Narumi–Katayama pada graf non-commuting yang terkait dengan grup aljabar tertentu, khususnya berfokus pada grup U_{6n}. Indeks Narumi–Katayama, yang pertama kali diperkenalkan oleh Narumi dan Katayama pada tahun 1984, adalah indeks topologis berbasis derajat yang banyak digunakan dalam studi berbagai properti graf, termasuk penerapannya dalam kimia teoretis. Graf non-commuting, di mana dua elemen saling berhubungan jika dan hanya jika mereka tidak komutatif, telah menjadi objek studi yang menarik dalam beberapa tahun terakhir. Sejauh yang kami ketahui, ini adalah studi pertama yang menghasilkan ekspresi bentuk tertutup untuk indeks First Zagreb dan Narumi–Katayama pada graf non-commuting dari grup U_{6n}. Berdasarkan penelitian sebelumnya tentang indeks detour dan konektivitas eksentrik pada graf Γ(U_{6n}), karya ini memberikan kontribusi baru dengan menghasilkan rumus umum yang dapat diterapkan pada kelas grup non-komutatif yang lebih luas. Berbeda dengan studi sebelumnya yang berfokus pada graf commuting atau coprime, penelitian ini secara khusus membahas struktur dan perhitungan indeks pada graf non-commuting dalam konteks teori grup. Hasil penelitian ini memberikan wawasan teoretis baru dalam teori graf aljabar dengan menghubungkan indeks berbasis derajat dengan struktur internal grup non-abelian. Diharapkan, temuan ini akan memperluas pemahaman tentang properti topologis graf non-commuting dan memberikan koneksi yang berharga untuk penerapan kimia.Kata Kunci: Graf non-commuting; Indeks Zagreb Pertama; Indeks Narumi–Katayama; Topologi graf; Struktur grup. 2020MSC: 05C25, 05C09, 20D60, 05C75.
Super (a,d)-P_2⨀P_m-Antimagic Total Labeling of Corona Product of Two Paths Yatin, Bela Zainun; Awanis, Zata Yumni; Wardhana, I Gede Adhitya Wisnu
JTAM (Jurnal Teori dan Aplikasi Matematika) Vol 8, No 2 (2024): April
Publisher : Universitas Muhammadiyah Mataram

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.31764/jtam.v8i2.20065

Abstract

Graph labeling involves mapping the elements of a graph (edges and vertices) to a set of positive integers. The concept of an anti-magic super outer labeling (a,d)-H pertains to assigning labels to the vertices and edges of a graph using natural numbers {1,2,3,...,p+q}. The weights of the outer labels H form an arithmetic sequence {a,a+d,a+2d,...,a+(k-1)d}, where 'a' represents the first term, 'd' is the common difference, and 'k' denotes the total number of outer labels, with the smallest label assigned to a vertex. This study investigates the super (a,d)-P_2⨀P_m-antimagic total labeling of the corona product P_n⨀P_m, where n and m are both greater than or equal to 3. We define the labeling functions for vertices and edges based on the partitioning of labels into three subsets. Using k-balanced and (k,δ)-anti balanced multisets, we demonstrate that for m being odd, P_n⨀P_m is super (9m^2 n+4mn+m-n+3,1)-P_2 ⨀▒P_(m ) -antimagic, and for m being even, P_n⨀P_m is super (9m^2 n+4mn+m-2n+5,3)-P_2 ⨀▒P_(m ) -antimagic. The labeling scheme is illustrated through examples. For the case when m is odd, an antimagic total labeling of P_3 ⨀▒P_3    forms a super (282,1)- P_2 ⨀▒P_(3 )  -antimagic labeling. In the case of even m, an antimagic total labeling of P_3 ⨀▒P_(4 ) results in a super (483,3)- P_2 ⨀▒P_(4 )  -antimagic labeling. Both of these examples provide insights into the antimagic properties of corona products.
ABC Index Analysis: Physical Properties of Prenylated Xanthone Pratiwi, Lia Fitta; Mufarrihati, Ardelia; Dharmayani, Ni Komang Tri; Wardhana, I Gede Adhitya Wisnu
JURNAL DIFERENSIAL Vol 7 No 2 (2025): November 2025
Publisher : Program Studi Matematika, Universitas Nusa Cendana

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.35508/jd.v7i2.20418

Abstract

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
Analisis Hubungan Nilai Indeks Zagreb dengan Titik Didih dan Kestabilan Senyawa Kerangka Dasar Calamenene Sabil, M. Ibnu; Miftahurrahman, Miftahurrahman; Wardhana, I Gede Adhitya Wisnu; Dharmayani, Ni Komang Tri
JURNAL DIFERENSIAL Vol 7 No 2 (2025): November 2025
Publisher : Program Studi Matematika, Universitas Nusa Cendana

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.35508/jd.v7i2.20431

Abstract

A new compound, dysoxyphenol, isolated from Dysoxylum densiflorum, has acalamenene-based framework. The structure of dysoxyphenol is classified as a sesquiterpene. Thisstudy discusses the relationship between Zagreb Index values and the physical properties andstability of two compounds: calamenene and dysoxyphenol. The Zagreb Index, consisting of thefirst and second indices, is used to measure the structural stability of a graph through the sum ofsquared vertex degrees. The objective of this analysis is to determine the Zagreb Index values forcalamenene and dysoxyphenol, to measure the boiling points and stability of the compounds. Thisanalysis was carried out by examining the structures of calamenene and dysoxyphenol. The obtainedstructures were then analyzed graphically. The results of this study show that the first and secondZagreb Index values for calamenene are 262 and 340, which are smaller compared to the first andsecond Zagreb Index values for dysoxyphenol, which are 264 and 350, respectively. This indicatesthat calamenene has a lower boiling point than dysoxyphenol and is more stable than dysoxyphenol.
Indeks Sombor dari Graf Koprima Prima untuk Grup Bilangan Bulat Modulo Abdurahim, Abdurahim; Satriyantara, Rio; Maulana, Fariz; Wardhana, I Gede Adhitya Wisnu; Robbaniyyah, Nuzla Af’idatur
JURNAL DIFERENSIAL Vol 7 No 2 (2025): November 2025
Publisher : Program Studi Matematika, Universitas Nusa Cendana

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.35508/jd.v7i2.21120

Abstract

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.
Submodul Prima Lemah dan Submodul Hampir Prima Pada Z‐modul M_2x2 (Z_9) Wardhana, I Gede Adhitya Wisnu; Switrayni, Ni Wayan; Aini, Qurratul
Eigen Mathematics Journal Vol 1 No 1: Vol 1 No 1 Juni 2018
Publisher : University of Mataram

Show Abstract | Download Original | Original Source | Check in Google Scholar | Full PDF (218.633 KB) | DOI: 10.29303/emj.v1i1.6

Abstract

Prime submodule is the abstraction to module theory of prime ideal in ring theory.  A proper submodule N of an R-module M is called prime submodule if for all r in R and m in M such that rm in N implies r in (N:M) or m in N.  Prime submodule also generalized into weakly prime submodule and almost prime submodule.  This study deal with particular cases of both of them in Z-module M_2x2(Z_9), the three submodules are equivalent in case of non-zero submodule.
TOPOLOGY INDEX OF THE COPRIME GRAPH FOR DIHEDRAL GROUP OF PRIME POWER ORDER Gayatri, Marena Rahayu; Fadhilah, Rifdah; Lestari, Sahin Two; Pratiwi, Lia Fitta; Abdurahim, Abdurahim; Wardhana, I Gede Adhitya Wisnu
JURNAL DIFERENSIAL Vol 5 No 2 (2023): November 2023
Publisher : Program Studi Matematika, Universitas Nusa Cendana

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.35508/jd.v5i2.12462

Abstract

In the field of molecular chemistry, graph theory is utilized to represent the structure of a molecule, where the set of nodes corresponds to its chemical elements and the set of edges represents the bonds within the chemical molecule. Graph theory, a mathematical discipline, finds application in various domains, one of which is group representation. This research will delve into the topic of the topological indices of the coprime graph of dihedral groups. The methodology employed involves reviewing several references related to dihedral groups, coprime graphs, and topological indices. This study yields results in the form of Harmonic index, Harary index, first Zagreb index, Gutman index, and Wiener index.
Co-Authors @ Ismail, Ghazali Semil A.A. Ketut Agung Cahyawan W Abdul Gazir Syarifudin Abdullah, Umar Abdurahim, Abdurahim Ade Candra, Ade Adelia Adelia Adelia Adelia, Adelia Aenan Salsabila Afdhaluzzikri, M. Ahmadil Hamdi Albaracin, Jimboy R. Alimon, Nur Idayu Ambar, Jinan Angamuthu, Manimaran Anisa Agustina Anisa Agustina, Anisa Apriliana, Haeva Ardana, Alfian Putra Arisanti, Devia Arzaki Zaget Oasis Asmarani, Evi Yuniartika Aulia, Sita Armi Awanis, Zatta Yumni Ayes Malona Siboro Ayes Malona Siboro Ayes Malona Siboro Baiq Desy Aniska Prayanti Baiq Rika Ayu Febrilia Beni Nungroho Sudiantoro Biswas, Hena Rani Borisman Bertinegara Dara Purnamasari Dara Puspita Anggraeni Devia Arisanti Dewi, Putu Kartika Dina Eka Putri Dwi Noorma Putri Elfiyanti, Gustina Emmy Yuanita Evi Yunartika Asmarani Evi Yuniartika Asmarani Evi Yuniartika Asmarani Evi Yuniartika Asmarani Fadhilah, Rifdah Farwan, Farwan Fathul Maulina Wahidah Febrilia, Baiq Rika Ayu Gambo, Ibrahim Gayatri, Marena Rahayu Ghazali Semil @ Ismail Ghoffari, Lalu Hasan Gilman, M. Afdhol Graha, Syifa Salsabila Satya Hapsari, Mufidatul Ghina Haryati, Ida Hidayat, Muhammad Ahsan Hijriati, Naimah Hisan, Khairatun Husni, Muhammad Naoval Ida Rohani Ilham Ilham Ilham Ilham Indrawadi, Dimas Intan Muchtadi Alamsyah Intan Nadilah Irwansyah Irwansyah Irwansyah Irwansyah Jurnal Pepadu Karang, Gusti Yogananda Laila Hayati Lailia Awalushaumi Lalu Hasan Ghoffari Lalu Riski Wirendra Putra Lalu Riski Wirendra Putra Lestari, Dia Lestari, Sahin Two Luzianawati, Luzianawati M Fauzul M. Afdhol Gilman Ma'wa, Jannatul Malik, Deny Putra MAMIKA UJIANITA ROMDHINI Mamika Ujianita Romdhini Mamika Ujianita Romdhini, Mamika Ujianita Maria Ulfa Masriani Masriani Masriani Masriani Maulana, Fariz Maulana, Muklas Maulani Rizqi Maulida Septiyana MAXRIZAL Miftahurrahman, Miftahurrahman Misuki, Wahyu Ulyafandhie Mufarrihati, Ardelia Muhammad Naoval Husni Muhammad Rijal Alfian Muklas Maulana Munawara Putia Musyarrofah, Sefti Fajriatul Nghiem, Nguyen Dang Hoa Ni Wayan Switrayni Ni Wayan Switrayni Ni Wayan Switrayni Nikken Prima Puspita Ningsih, Baiq Nila Sari Nur Asmita Purnamasari Nurhabibah Nurhabibah Nurhabibah Nurhabibah Nurhabibah Nurhabibah Nuzla Af'idatur Robbaniyyah Oasis, Arzaki Zaget Pradana, Satriawan Pratama, Rendi Bahtiar Pratiwi, Lia Fitta Prof. Dr.I Nengah Suparta,M.Si . PUDJI ASTUTI Purnamasari, Dara Putia, Munawara Putra, Lalu Riski Wirendra Putri Kurnia Chairunnisa Putri, Syaftirridho Putu Kartika Dewi Qudrani , Rabbelia Tri Qudrani, Rabbelia Qurratul Aini Qurratul Aini Qurratul Aini Ramdani, Dewi Santri Rendi Bahtiar Pratama Rina Juliana Rina Juliana Rio Satriyantara Robbaniyyah, Nuzla Af’idatur Rohani, Ida Rohiana, Siti Indah Sabil, M. Ibnu Sahin Two Lestari Sahin Two Lestari Salsabila, Aenan Salwa Salwa Salwa Salwa Salwa Salwa Salwa Salwa Salwa Salwa Santi, Laila Maya Sari, Mutia Nofita Sarmin, Nor Haniza Satriawan, Didit Semil @ Ismail, Ghazali Semil Ismail, Ghazali Shaumi, Nurina Fadlila Siboro, Ayes Malona Siti Raudhatul Kamali Sudiantoro, Beni Nungroho Sudirman Sudirman Surya Hadi Suwastika, Erma Syafitri, Hanna Syaftirridho Putri Syawaludin, Muhammad Khair Tri Dharmayani, Ni Komang Tri Maryono Rusadi Ubaidillah, Moch Rafi Zarkasy Wahidah, Fathul Maulina Widiastuti, Ratna Sari Yatin, Bela Zainun Zata Yumni Awanis Zata Yumni Awanis Zata Yumni Awaris