The First Zagreb Index, The Wiener Index, and The Gutman Index of The Power of Dihedral Group

Asmarani, Evi Yuniartika; Lestari, Sahin Two; Purnamasari, Dara; Syarifudin, Abdul Gazir; Salwa, Salwa; Wardhana, I Gede Adhitya Wisnu

doi:10.18860/ca.v7i4.16991

Cited by 4 publications

(2 citation statements)

References 10 publications

(13 reference statements)

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“…Penerapan teori graf lebih lanjut pada ilmu matematika adalah dalam menentukan indeks topologi. Penelitian tentang indeks topologi telah dilakukan oleh [6][7][8] yang membahas indeks harmonik, indeks Zagreb, indeks Weiner dan indeks Gutman dari graf koprima maupun graf pangkat dari grup dihedral, grup quternion yang diperumum atau grup bilangan bulat modulo.…”

Section: Pendahuluanunclassified

Topology Index of the Coprime Graph for Dihedral Group of Prime Power Order

Gayatri,

Fadhilah,

Lestari

et al. 2023

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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.

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Section: Pendahuluanunclassified

Topology Index of the Coprime Graph for Dihedral Group of Prime Power Order

Gayatri,

Fadhilah,

Lestari

et al. 2023

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“…Commuting graphs, on the other hand, investigate the relationships between group elements that commute with each other. These diverse graph representations provide valuable insights into different aspects of group theory and its applications in mathematics [11]- [19].…”

Section: Introductionmentioning

confidence: 99%

The Structures of Non-Coprime Graphs for Finite Groups from Dihedral Groups with Regular Composite Orders

Aulia,

Wardhana,

Irwansyah

et al. 2023

InPrime:Ind.Jour.Pure.Applied.Math

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For any finite group, the non-coprime graph of the group is a graph with vertices consisting of all non-identity elements of the group. Two different vertices are considered adjacent if their orders are not coprime, meaning their greatest common divisor (gcd) is not equal to one. Misuki provides the structure of the non-coprime graph for the dihedral group when the order is a prime power. We establish a more general property for cases where the order of the group is a regular composite number, discovering that the structure of the non-coprime graph for a dihedral group can be partitioned into some number of complete subgraphs.Keywords: dihedral group; disjoint subgraph; non-coprime graph. AbstrakUntuk sebarang grup hingga, graf non-coprime dari grup tersebut adalah graf dengan simpul yang terdiri dari semua elemen non-identitas dari grup tersebut. Dua simpul yang berbeda dianggap bertetangga jika orde mereka tidak saling prima, artinya pembagi terbesar bersama (gcd) mereka tidak sama dengan satu. Misuki memberikan struktur graf non-coprime untuk grup dihedral ketika ordenya adalah pangkat prima. Pada studi ini didapatkan sifat yang lebih umum ketika orde grup adalah bilangan komposit biasa. Didapatkan juga bahwa struktur graf non-coprime dari grup dihedral dapat dipartisi menjadi beberapa subgraf lengkap.Kata Kunci: graf non-koprima; grup dihedral; subgraph disjoin. 2020MSC: 05C25, 05C69, 20D60.

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Machine Learning Driven Exploration of Energies and Generalization of Topological Indices for the Fuzzy Conjugate Graph of Dihedral Group

Umar Mirza,

Anjum,

Fahad Alrasheedi

et al. 2024

IEEE Access

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This study introduces a ground-breaking approach to analyzing dihedral groups through the lens of fuzzy graph theory, significantly enhancing computational efficiency in group theory. By extending the fuzzy topological indices to polynomial forms, this research drastically reduces the calculation time for these indices from hours to mere seconds. A notable feature of this work is the innovative use of polynomial regression, a machine learning technique, to generate polynomials for adjacency and degree matrices within the fuzzy conjugate graph of a dihedral group. This method not only simplifies calculations but also incorporates an error analysis component, ensuring accuracy and reliability. The integration of fuzzy graph theory with polynomial regression in this context is a pioneering step, offering valuable insights into the structural attributes of dihedral groups. This research goes beyond traditional methods, highlighting the effectiveness of machine learning in deciphering complex patterns in group theory. The findings and techniques presented hold great promise for future applications in graph theory and group theory, offering a novel perspective for understanding and analyzing intricate graph structures. This study stands as a significant contribution to the field, potentially revolutionizing the way complex mathematical problems are approached and solved.

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The First Zagreb Index, The Wiener Index, and The Gutman Index of The Power of Dihedral Group

Cited by 4 publications

References 10 publications

Topology Index of the Coprime Graph for Dihedral Group of Prime Power Order

Topology Index of the Coprime Graph for Dihedral Group of Prime Power Order

The Structures of Non-Coprime Graphs for Finite Groups from Dihedral Groups with Regular Composite Orders

Machine Learning Driven Exploration of Energies and Generalization of Topological Indices for the Fuzzy Conjugate Graph of Dihedral Group

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