Statistical Bounds for the Energy of Graphs

Document Type : Original Scientific Paper

Authors

‎Department of Mathematics‎, ‎‎Tafresh University, Tafresh 39518-79611‎, ‎I‎. ‎R‎. ‎Iran

10.22052/mir.2025.257557.1538

Abstract

‎This paper proposes several new statistical bounds for graph energy derived from the eigenvalues of the adjacency matrix‎. ‎Using inequalities involving the arithmetic‎, ‎geometric‎, ‎and generalized means‎, ‎along with variance and standard deviation‎, ‎we establish both upper and lower bounds for $E(G)$‎. ‎These statistical bounds capture not only mean relationships but also eigenvalue variability‎, ‎offering more flexible and accurate estimates than conventional deterministic inequalities‎. ‎The approach integrates tools from inequality theory and spectral graph theory‎, ‎with applying weighted means and Jensen-type inequalities‎. ‎We also conjecture based on numerical evidence that the energy-to-geometric mean ratio converges to a constant value for large Erd\"{o}s-R'{e}nyi random graphs‎. ‎A detailed analysis of path graphs demonstrates the effectiveness of the proposed bounds‎, ‎offering improved estimates‎.

Keywords

Main Subjects

References

[1] S. Akbari, A. H. Ghodrati, M. A. Hosseinzadeh, Some lower bounds for the energy of graphs, Linear Algebra Appl. 591 (2020) 205-214, https://doi.org/10.1016/j.laa.2020.01.001.
[2] E. Estrada and M. Benzi, What is the meaning of the graph energy after all?, Discrete Appl. Math. 230 (2017) 71-77,
https://doi.org/10.1016/j.dam.2017.06.007.
[3] A. Jahanbani, Upper bounds for the energy of graphs, MATCH Commun. Math. Comput. Chem. 79 (2018) 275-286.
[4] H. Barzegar and M. Mohammadi, Bounds for Sombor index using topological and statistical indices, J. Disc. Math. Appl. 10 (2025) 61-85, https://doi.org/10.22061/JDMA.2025.11494.1107.
[5] G. Sridhara, M. R. Rajesh Kanna and H. L. Parashivamurthy, Energy of graphs and its new bounds, South East Asian J. Math. Math. Sci. 18 (2022) 161-170, https://doi.org/10.56827/SEAJMMS.2022.1802.15.
[6] H. Barzegar, Lower and upper bounds between energy, laplacian energy, and Sombor index of some graphs, Iranian J. Math. Chem. 16 (2025) 33-38, https://doi.org/ 10.22052/IJMC.2024.254674.1850.
[7] M. Khan, A new notion of energy of digraphs, Iranian J. Math. Chem. 12 (2021) 111-125, https://doi.org/10.22052/IJMC.2020.224853.1496.
[8] J. M. Aldaz, Self-improvement of the inequality between arithmetic and geometric means, J. Math. Inequal. 3 (2009) 213-216.
[9] J. M. Aldaz, Comparison of differences between arithmetic and geometric, Tamkang J. Math. 42 (2011) 453-462, https://doi.org/10.5556/j.tkjm.42.2011.747.
[10] J. Liu, Y. Chen, D. Dimitrov and J. Chen, New bounds on the energy of graphs with self–loops, MATCH Commun. Math. Comput. Chem. 91 (2024) 779-796, https://doi.org/10.46793/match.91-3.779L.
[11] B. Rodin, Variance and the inequality of arithmatic and geometric means, J. Math. 47 (2017) 637-648, https://doi.org/10.1216/RMJ-2017-47-2-637.
[12] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004.
[13] S. Janson, T. Łuczak and A. Rucinski, Random Graphs, John Wiley & Sons, New York, 2000.
Mathematics Interdisciplinary Research
Volume 11, Issue 1
March 2026
Pages 31-42

Files

Share

How to cite

Statistics