Abstract
In this study, the Laplacian matrix concept for the power graph of a finite cyclic group is redefined by considering the block matrix structure. Then, with the help of the eigenvalues of the Laplacian matrix in question, the concept of Laplacian energy for the power graphs of finite cyclic groups was defined and introduced into the literature. In addition, boundary studies were carried out for the Laplacian energy in question using the concepts the trace of a matrix, the Cauchy-Schwarz inequality, the relationship between the arithmetic mean and geometric mean, and determinant. Later, various results were obtained for the Laplacian energy in question for cases where the order of a cyclic group is the positive integer power of a prime.
References
- [1] A. V. Kelarev, S. J. Quinn, “A Combinatorial Property and Power Graphs of Groups,” Contribibutions to General Algebra, vol. 12, pp. 229-235, 2000.
- [2] A. V. Kelarev, S. J. Quinn, “Directed Graphs and Combinatorial Properties of Semigroups,” Journal of Algebra, vol. 251, no. 1, pp. 16-26, 2002.
- [3] A. V. Kelarev, S. J. Quinn, “A Combinatorial Property and Power Graphs of Semigroups,” Commentationes Mathematicae Universitatis Carolinae, vol. 45, no. 1, pp. 1-7, 2004.
- [4] I. Chakrabarty, S. Ghosh, M. K. Sen, “Undirected Power Graphs of Semigroups,” Semigroup Forum, vol. 78, pp. 410-426, 2009.
- [5] P. J. Cameron, “The Power Graph of AFinite Group II.,” Journal of Group Theory, vol. 13, no. 6, pp. 779-783, 2010.
- [6] P. J. Cameron, S. Ghosh, “The Power Graph of A Finite Group,” Discrete Mathematics, vol. 311, no. 13, pp. 1220-1222, 2011.
- [7] S. Chattopadhyay, P. Panigrahi, F. Atik, “Spectral Radius of Power Graphs on Certain Finite Groups,” Indagationes Mathematicae, vol.29, no. 2, pp. 730-737, 2018.
- [8] I. Gutman, “The Energy of Graph,” Berichteder Mathematisch Statistischen Sektion im Forschungszentrum Graz, vol. 103, pp. 1-22, 1978.
- [9] I. Gutman, B. Zhou, “Laplacian Energy of AGraph,” Linear Algebra and its Applications, vol. 414, no. 1, pp. 29-37, 2006.
Abstract
References
- [1] A. V. Kelarev, S. J. Quinn, “A Combinatorial Property and Power Graphs of Groups,” Contribibutions to General Algebra, vol. 12, pp. 229-235, 2000.
- [2] A. V. Kelarev, S. J. Quinn, “Directed Graphs and Combinatorial Properties of Semigroups,” Journal of Algebra, vol. 251, no. 1, pp. 16-26, 2002.
- [3] A. V. Kelarev, S. J. Quinn, “A Combinatorial Property and Power Graphs of Semigroups,” Commentationes Mathematicae Universitatis Carolinae, vol. 45, no. 1, pp. 1-7, 2004.
- [4] I. Chakrabarty, S. Ghosh, M. K. Sen, “Undirected Power Graphs of Semigroups,” Semigroup Forum, vol. 78, pp. 410-426, 2009.
- [5] P. J. Cameron, “The Power Graph of AFinite Group II.,” Journal of Group Theory, vol. 13, no. 6, pp. 779-783, 2010.
- [6] P. J. Cameron, S. Ghosh, “The Power Graph of A Finite Group,” Discrete Mathematics, vol. 311, no. 13, pp. 1220-1222, 2011.
- [7] S. Chattopadhyay, P. Panigrahi, F. Atik, “Spectral Radius of Power Graphs on Certain Finite Groups,” Indagationes Mathematicae, vol.29, no. 2, pp. 730-737, 2018.
- [8] I. Gutman, “The Energy of Graph,” Berichteder Mathematisch Statistischen Sektion im Forschungszentrum Graz, vol. 103, pp. 1-22, 1978.
- [9] I. Gutman, B. Zhou, “Laplacian Energy of AGraph,” Linear Algebra and its Applications, vol. 414, no. 1, pp. 29-37, 2006.
Details
| Primary Language | English |
|---|---|
| Subjects | Algebra and Number Theory, Pure Mathematics (Other) |
| Journal Section | Research Articles |
| Authors | |
| Early Pub Date | April 26, 2024 |
| Publication Date | April 30, 2024 |
| Submission Date | October 2, 2023 |
| Acceptance Date | January 31, 2024 |
| Published in Issue | Year 2024 Volume: 28 Issue: 2 |
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