Quantization of Sombor Energy for ‎Complete ‎Graphs with‎ ‎Self-Loops of‎ ‎Large Size

Document Type : Research Paper

Authors

School of Mathematical Sciences, Universiti Sains Malaysia, Malaysia

Abstract

‎A self-loop graph $G_S$ is a simple graph $G$ obtained by attaching loops at $S \subseteq V(G).$ To such $G_S$ an Euclidean metric function is assigned to its vertices‎, ‎forming the so-called Sombor matrix‎. ‎In this paper‎, ‎we derive two summation formulas for the spectrum of the Sombor matrix associated with $G_S,$ for which a Forgotten-like index arises‎. ‎We explicitly study the Sombor energy $\cE_{SO}$ of complete graphs with self-loops $(K_n)_S,$ as the sum of the absolute value of the difference of its Sombor eigenvalues and an averaged trace‎. ‎The behavior of this energy and its change for a large number of vertices $n$ and loops $\sigma$ is then studied‎. ‎Surprisingly‎, ‎the constant $4\sqrt{2}$ is obtained repeatedly in several scenarios‎, ‎yielding a quantization of the energy change of 1 loop for large $n$ and $\sigma$‎.
‎Finally‎, ‎we provide a McClelland-type and determinantal-type upper and lower bounds for $\cE_{SO}(G_S),$ which generalizes several bounds in the literature‎.

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References

[1] I. Gutman, The energy of a graph, Ber. Math.— Statist. Sekt. Forschungsz. Graz 103 (1978) 1–22.
[2] M. Cavers, S. Fallat and S. Kirkland, On the normalized Laplacian energy and general Randic index R􀀀1 of graphs, Linear Algebra Appl. 433 (2010) 172–190, https://doi.org/10.1016/j.laa.2010.02.002.
[3] I. Gutman, I. Redžepovic, B. Furtula and A. Sahal, Energy of graphs with self-loops, MATCH Commun. Math. Comput. Chem. 87 (3) (2022) 645–652, https://doi.org/10.46793/match.87-3.645G.
[4] I. M. Jovanovic, E. Zogic and E. Glogic, On the conjecture related to the energy of graphs with self-loops, MATCH Commun. Math. Comput. Chem. 89 (2023) 479–488, https://doi.org/10.46793/match.89-2.479J.
[5] S. Akbari, H. Al Menderj, M. H. Ang, J. Lim and Z. C. Ng, Some results on spectrum and energy of graphs with loops, Bull. Malays. Math. Sci. Soc. 46 (2023) p. 94, https://doi.org/10.1007/s40840-023-01489-z.
[6] I. Gutman, Spectrum and energy of the Sombor matrix, Military Tech. Courier 69 (3) (2021) 551–561, https://doi.org/10.5937/vojtehg69-31995.
[7] K. J. Gowtham and S. N. Narasimha, On Sombor energy of graphs, Nanosystems: Phys. Chem. Math. 12 (4) (2021) 411–417.
[8] Z. Lin, T. Zhou and L. Miao, On the spectral radius, energy and Estrada index of the Sombor matrix of graphs, Trans. Comb. 12 (4) (2023) 191–205, https://doi.org/10.22108/TOC.2022.127710.1827.
[9] N. Biggs, Algebraic Graph Theory, Cambridge University Press, 1993.
[10] S. R. Garcia and R. A. Horn, A Second Course in Linear Algebra, Cambridge University Press, 2017.
[11] N. Ghanbari, On the Sombor characteristic polynomial and Sombor energy of a graph, Comput. Appl. Math. 41 (6) (2022) p. 242, https://doi.org/10.1007/s40314-022-01957-5.
[12] B. J. McClelland, Properties of the latent roots of a matrix: the estimation of $\pi$-electron energies, J. Chem. Phys. 54 (2) (1971) 640–643, https://doi.org/10.1063/1.1674889.
[13] K. C. Das, I. Gutman, I. Milovanovic, E. Milovanovic and B. Furtula, Degree-based energies of graphs, Linear Algebra Appl. 554 (2018) 185–204, https://doi.org/10.1016/j.laa.2018.05.027.
[14] I. Gutman and L. Pavlovic, The energy of some graphs with large number of edges, Bull. Cl. Sci. Math. Nat. Sci. Math. 118 (1999) 35–50.
[15] I. Gutman, Topological studies on heteroconjugated molecules, Theoret. Chim. Acta 50 (1979) 287–297, https://doi.org/10.1007/BF00551336.
[16] I. Gutman, Topological studies on heteroconjugated molecules. VI. Alternant systems with two heteroatoms, Z. Naturforsch. 45a (1990) 1085–1089, https://doi.org/10.1515/zna-1990-9-1005.
[17] R. B. Mallion, A. J. Schwenk and N. Trinajstic, Graphical study of heteroconjugated molecules, Croat. Chem. Acta 46 (3) (1974) 171–182.
[18] R. B. Mallion, N. Trinajstic and A. J. Schwenk, Graph theory in Chemistry-Generalisation of Sachs’ Formula, Z. Naturforsch. 29a (1974) 1481–1484, https://doi.org/10.1515/zna-1974-1016.
Iranian Journal of Mathematical Chemistry
Volume 14, Issue 4
December 2023
Pages 225-241

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