Neighbors degree sum energy of commuting and non-commuting graphs for dihedral groups. (English) Zbl 1533.05170
Summary: The neighbors degree sum (\(NDS\)) energy of a graph is determined by the sum of its absolute eigenvalues from its corresponding neighbors degree sum matrix. The non-diagonal entries of \(NDS\)-matrix are the summation of the degree of two adjacent vertices, or it is zero for non-adjacent vertices, whereas for the diagonal entries are the negative of the square of vertex degree. This study presents the formulas of neighbors degree sum energies of commuting and non-commuting graphs for dihedral groups of order \(2n\), \(D_{2n}\), for two cases – odd and even \(n\). The results in this paper comply with the well known fact that energy of a graph is neither an odd integer nor a square root of an odd integer.
MSC:
| 05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
Keywords:
commuting graph; non-commuting graph; dihedral group; neighbors degree sum matrix; energy of a graphSoftware:
MapleReferences:
| [1] | A. Abdollahi, S. Akbari & H. Maimani (2006). Non-commuting graph of a group. Journal of Algebra, 298(2), 468-492. https://doi.org/10.1016/j.jalgebra.2006.02.015. · Zbl 1105.20016 · doi:10.1016/j.jalgebra.2006.02.015 |
| [2] | M. Aschbacher (2000). Finite Group Theory. Cambridge University Press, Cambridge, United Kingdom. · Zbl 0997.20001 |
| [3] | R. Bapat & S. Pati (2004). Energy of a graph is never an odd integer. Bulletin of Kerala Mathe-matics Association, 1, 129-132. · Zbl 1257.05082 |
| [4] | K. Bhat & G. Sudhakara (2018). Commuting graphs and their generalized complements. Malaysian Journal of Mathematical Sciences, 12(1), 63-84. · Zbl 07148982 |
| [5] | R. Brauer & K. Fowler (1955). On groups of even order. Annals of Mathematics, 62(3), 565-583. https://doi.org/10.2307/1970080. · Zbl 0067.01004 · doi:10.2307/1970080 |
| [6] | D. Bundy (2006). The connectivity of commuting graphs. Journal of Combinatorial Theory, Series A, 113(6), 995-1007. https://doi.org/10.1016/j.jcta.2005.09.003. · Zbl 1098.05046 · doi:10.1016/j.jcta.2005.09.003 |
| [7] | B. W. Char, K. O. Geddes, G. H. Gonnet, B. L. Leong, M. B. Monagan & S. Watt (2013). Maple V Library Reference Manual. Springer Science and Business Media, Berlin, Germany. |
| [8] | S. Chattopadhyay, P. Panigrahi & F. Atik (2018). Spectral radius of power graphs on certain finite groups. Indagationes Mathematicae, 29(2), 730-737. https://doi.org/10.1016/j.indag. 2017.12.002. · Zbl 1382.05042 · doi:10.1016/j.indag.2017.12.002 |
| [9] | J. Dutta & R. Nath (2017). Spectrum of commuting graphs of some classes of finite groups. Matematika, 33(1), 87-95. |
| [10] | W. N. T. Fasfous & R. K. Nath. Spectrum and energy of non-commuting graphs of finite groups. Available at https://doi.org/10.48550/arXiv.2002.10146. · doi:10.48550/arXiv.2002.10146 |
| [11] | H. Ganie & Y. Shang (2022). On the spectral radius and energy of signless laplacian matrix of digraphs. Heliyon, 8(3), 1-6. https://doi.org/10.1016/j.heliyon.2022.e09186. · doi:10.1016/j.heliyon.2022.e09186 |
| [12] | F. R. Gantmacher (1959). The Theory of Matrices. Chelsea Publishing Company, New York. · Zbl 0085.01001 |
| [13] | I. Gutman (1978). The energy of graph. Ber. Math. Statist. Sekt. Forschungszenturm Graz., 103, 1-22. · Zbl 0402.05040 |
| [14] | M. Hakimi-Nezhaad & A. R. Ashrafi (2015). Some types of spectral distances between a hypercube and its complement and line graph. Malaysian Journal of Mathematical Sciences, 9(1), 145-159. |
| [15] | R. Horn & C. Johnson (1985). Matrix Analysis. Cambridge University Press, Cambridge, United Kingdom. · Zbl 0576.15001 |
| [16] | H. S. B. . R. B. Jummannaver (2021). Neighbors degree sum energy of graph. Journal of Applied Mathematics and Computing, 67, 579â603. https://doi.org/10.1007/s12190-020-01480-y. · Zbl 1487.05077 · doi:10.1007/s12190-020-01480-y |
| [17] | S. Kasim & A. Nawawi (2018). On the energy of commuting graph in symplectic group. AIP Conference Proceedings, 1974, Article ID 030022. https://doi.org/10.1063/1.5041666. · doi:10.1063/1.5041666 |
| [18] | S. Kasim & A. Nawawi (2021). On diameter of subgraphs of commuting graph in symplectic group for elements of order three. Sains Malaysiana, 50(2), 549-557. http://dx.doi.org/10. 17576/jsm-2021-5002-25. · doi:10.17576/jsm-2021-5002-25 |
| [19] | S. M. S. Khasraw, I. D. Ali & R. R. Haji (2020). On the non-commuting graph of dihedral group. Electronic Journal of Graph Theory and Applications, 8(2), 233-239. http://dx.doi.org/ 10.5614/ejgta.2020.8.2.3. · Zbl 1468.05058 · doi:10.5614/ejgta.2020.8.2.3 |
| [20] | R. Mahmoud, N. H. Sarmin & A. Erfanian (2017). On the energy of non-commuting graph of dihedral groups. AIP Conference Proceedings, 1830, Article ID 070011. https://doi.org/10. 1063/1.4980960. · doi:10.1063/1.4980960 |
| [21] | A. Nawawi (2013). On Commuting Graphs for Elements of Order 3 in Symmetric Groups, PhD Thesis. University of Manchester, United Kingdom. |
| [22] | A. Nawawi, S. K. S. Husain & M. R. K. Ariffin (2019). Commuting graphs, C(G,X)in symmetric groups sym(n) and its connectivity. Symmetry, 11(9), Article ID 1178. https: //doi.org/10.3390/sym11091178. · doi:10.3390/sym11091178 |
| [23] | A. Nawawi & P. Rowley (2015). On commuting graphs for elements of order 3 in symmetric groups. The Electronic Journal of Combinatorics, 22(1), 1-12. https://doi.org/10.37236/2362. · Zbl 1307.05111 · doi:10.37236/2362 |
| [24] | S. Pirzada & I. Gutman (2008). Energy of a graph is never the square root of an odd inte-ger. Applicable Analysis and Discrete Mathematics, 2(1), 118-121. http://www.jstor.org/stable/ 43666800. · Zbl 1199.05236 |
| [25] | H. Ramane & S. Shinde (2017). Degree exponent polynomial of graphs obtained by some graph operations. Electronic Notes in Discrete Mathematic, 63, 161-168. https://doi.org/10. 1016/j.endm.2017.11.010. · Zbl 1383.05275 · doi:10.1016/j.endm.2017.11.010 |
| [26] | M. U. Romdhini & A. Nawawi (2022b). Maximum and minimum degree energy of com-muting graph for dihedral groups. Sains Malaysiana, 51(12), 4145-4151. http://doi.org/10. 17576/jsm-2022-5112-21. · doi:10.17576/jsm-2022-5112-21 |
| [27] | M. U. Romdhini & A. Nawawi (2022c). Degree sum energy of non-commuting graph for dihedral groups. Malaysian Journal of Science, 41(SP1), 34-39. https://doi.org/10.22452/mjs. sp2022no.1.5. · doi:10.22452/mjs.sp2022no.1.5 |
| [28] | M. U. Romdhini, A. Nawawi & C. Y. Chen (2022a). Degree exponent sum energy of com-muting graph for dihedral groups. Malaysian Journal of Science, 41(SP1), 40-46. https: //doi.org/10.22452/mjs.sp2022no1.6. · doi:10.22452/mjs.sp2022no1.6 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
