Topological Indices of Total Graph and Zero Divisor Graph of Commutative Ring: A Polynomial Approach

Mondal, Sourav; Imran, Muhammad; De, Nilanjan; Pal, Anita

doi:10.1155/2023/6815657

Cited by 4 publications

(3 citation statements)

References 49 publications

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“…These indices are algebraic quantities that correlate the chemical structure with physical characteristics and can be used to determine properties like chemical activity, thermodynamic properties, and biological activity [16]. Topological indices are also used in the study of quantita-tive structure-activity relationships (QSAR) and quantitative structure-property relationships (QSPR) to correlate molecular structure with different properties and activities [17]. They have been applied in drug design and development to analyze the physical characteristics, chemical reactivity, and biological activity of chemical structures [18].…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 99%

“…FIGURE 7. absolute error between exact adjacency energies and approximated adjacency energies by with equation(17).…”

mentioning

confidence: 99%

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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“…The vertex degree of vertex 𝑣 on graph 𝐺 is denoted by 𝑑𝑒𝑔(𝑣) is the number of vertices adjacent to 𝑣 in graph 𝐺 (Akbari et al, 2009). Several researchers have researched total graphs, including (Anderson & Badawi, 2008), (Akbari et al, 2009), and (Mondal et al, 2023). Anderson and Weber (2018) investigated zero-divisor graphs of 𝑅 , where 𝑅 contains no identities.…”

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confidence: 99%

Algorithm for Constructing Total Graph of Commutative Ring

Meinawati,

Kurniawan,

Kurdhi

2024

JTAM

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Let R be a commutative ring. The total graph of R, denoted by TΓ(R) is a graph whose vertices are all elements of the ring R and every i,j∈R with i≠j, then i and j vertices are connected by edges if and only if i+j∈Z(R), where Z(R) is the set of zero-divisors in R with 0∈Z(R). Python programming is code that is easy to learn, read, understand, and helpful in explaining problems regarding graphs and algebra. In this paper, we determine an algorithm to construct the total graph of ring Z_n using Python. The research methods in this paper is a literature studies. The results generated by the algorithm can be utilized to observe the characteristic patterns displayed by the graph. Based on the algorithm’s constructed graph pattern, several properties of TΓ(Z_n ) can be inferred. For instance, if n is a prime number, then TΓ(Z_n ) is a disconnected graph. On the other hand, if n is a prime number and n≥3, then TΓ(Z_2n ) and TΓ(Z_4n ) is a connected graph, regular graph, Hamiltonian graph, and has a girth gr(TΓ(〖Z〗_n ))=3. In this paper we creating an algorithm to construct total graphs from commutative rings streamlines the construction process, enhances accessibility and utilization of total graphs, and supports parameter variation exploration and application in problem-solving.

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An Algebraic Approach of Topological Indices Connected with Finite Quasigroups

Nadeem,

Alam,

Ali

et al. 2024

Journal of Function Spaces

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In mathematical chemistry, the algebraic polynomial serves as essential for calculating the most accurate expressions of distance-based, degree-distance-based, and degree-based topological indices. The chemical reactivity of molecules, which includes their tendency to engage in particular chemical processes or go through particular reactions, can be predicted using topological indices. Considerable effort has been put into examining the many topological descriptors of simple graphs using ring structures and well-known groups instead of nonassociative algebras, quasigroups, and loops. Both finite quasigroups and loops are the generalizations of groups. In this article, we calculate topological descriptors and some well-known polynomials, M-polynomial, Hosoya’s polynomial, Schultz’s polynomial, and modified Schultz polynomial of finite relatively prime graphs of most orders connected with two classes of quasigroups and go through their graphical aspects.

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Topological Indices of Total Graph and Zero Divisor Graph of Commutative Ring: A Polynomial Approach

Cited by 4 publications

References 49 publications

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

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

Algorithm for Constructing Total Graph of Commutative Ring

An Algebraic Approach of Topological Indices Connected with Finite Quasigroups

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