Laplacian energy and Quasi-Laplacian energy for trees

Emrecan Erdem; Ayşe Dilek Maden; Masood Ur Rehman

doi:10.54286/ikjm.1760148

Laplacian energy and Quasi-Laplacian energy for trees

Abstract

For a simple graph $G$, the Laplacian matrix is denoted by $L(G)$ and the Signless Laplacian matrix is denoted by $Q(G)$. These matrices and their eigenvalues are widely used in many fields such as network measurement and chemical structures. Laplacian energy of a graph $G$ of order $n$ is defined as $LE(G)= \sum_{i=1}^n \ |\mu_i-\overline d |$, where $\mu_i$ is the $i$-th eigenvalue of Laplacian matrix of $G$, and $\overline d$ is average degree. Similarly, the Quasi-Laplacian energy of the graph $G$ of order $n$ is defined as $E_Q(G)= \sum_{i=1}^n \ \sigma_i^2$, where $\sigma_i$ is the $i$-th eigenvalue of Signless Laplacian matrix of $G$. The inequality $LE(P_n) \leq LE(T_n) \leq LE(S_n)$ was given as conjecture by Radenković and Gutman in 2007. This conjucture was proved by Trevisan et al. for trees with 3 diameters and by Rehman et al. for trees with 4 diameters. Our aim is to prove this conjecture for trees with larger diameters. In this study, we solved the above conjecture for some tree classes between 5 and 15 in diameter by the linear time algorithm for characteristic polynomials. It is also proved ${E_Q}\left( {{P_n}} \right) \le {E_Q}\left( {{T_n}} \right) \le {E_Q}\left( {{S_n}} \right) $ inequality, which is similar to Gutman's conjecture for the Quasi-Laplacian energy.

Keywords

References

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Details

Primary Language

English

Subjects

Combinatorics and Discrete Mathematics (Excl. Physical Combinatorics)

Journal Section

Research Article

Authors

Emrecan Erdem ^*
0000-0002-1183-5105
Türkiye

Ayşe Dilek Maden
0000-0001-7717-0241
Türkiye

Masood Ur Rehman
0000-0001-6343-0620
China

Publication Date

July 16, 2026

Submission Date

August 7, 2025

Acceptance Date

January 30, 2026

Published in Issue

Year 2026 Volume: 8 Number: 1

DOI

https://doi.org/10.54286/ikjm.1760148

IZ

https://izlik.org/JA48HG58GP

Cite

RIS / Bibtex

APA

Erdem, E., Maden, A. D., & Rehman, M. U. (2026). Laplacian energy and Quasi-Laplacian energy for trees. Ikonion Journal of Mathematics, 8(1), 1-15. https://doi.org/10.54286/ikjm.1760148

AMA

1.Erdem E, Maden AD, Rehman MU. Laplacian energy and Quasi-Laplacian energy for trees. ikjm. 2026;8(1):1-15. doi:10.54286/ikjm.1760148

Chicago

Erdem, Emrecan, Ayşe Dilek Maden, and Masood Ur Rehman. 2026. “Laplacian Energy and Quasi-Laplacian Energy for Trees”. Ikonion Journal of Mathematics 8 (1): 1-15. https://doi.org/10.54286/ikjm.1760148.

EndNote

Erdem E, Maden AD, Rehman MU (July 1, 2026) Laplacian energy and Quasi-Laplacian energy for trees. Ikonion Journal of Mathematics 8 1 1–15.

IEEE

[1]E. Erdem, A. D. Maden, and M. U. Rehman, “Laplacian energy and Quasi-Laplacian energy for trees”, ikjm, vol. 8, no. 1, pp. 1–15, July 2026, doi: 10.54286/ikjm.1760148.

ISNAD

Erdem, Emrecan - Maden, Ayşe Dilek - Rehman, Masood Ur. “Laplacian Energy and Quasi-Laplacian Energy for Trees”. Ikonion Journal of Mathematics 8/1 (July 1, 2026): 1-15. https://doi.org/10.54286/ikjm.1760148.

JAMA

1.Erdem E, Maden AD, Rehman MU. Laplacian energy and Quasi-Laplacian energy for trees. ikjm. 2026;8:1–15.

MLA

Erdem, Emrecan, et al. “Laplacian Energy and Quasi-Laplacian Energy for Trees”. Ikonion Journal of Mathematics, vol. 8, no. 1, July 2026, pp. 1-15, doi:10.54286/ikjm.1760148.

Vancouver

1.Emrecan Erdem, Ayşe Dilek Maden, Masood Ur Rehman. Laplacian energy and Quasi-Laplacian energy for trees. ikjm. 2026 Jul. 1;8(1):1-15. doi:10.54286/ikjm.1760148