Exact Formulas for the Energy, Laplacian Energy, and Signless Laplacian Energy of Line Graphs of Lintang Graphs

Authors

  • Fransiskus Fran Department Mathematics, Universitas Tanjungpura, Pontianak, 78124, Indonesia
  • Meliana Pasaribu Department Mathematics, Universitas Tanjungpura, Pontianak, 78124, Indonesia
  • Helmi Helmi Department Mathematics, Universitas Tanjungpura, Pontianak, 78124, Indonesia
  • Raventino Raventino Department Mathematics, Universitas Tanjungpura, Pontianak, 78124, Indonesia

DOI:

https://doi.org/10.30829/zero.v10i2.28777

Keywords:

Adjacency Matrix, Block Matrix, Eigen Value, Matrix Determinant, Spectrum Graph

Abstract

This study establishes exact formulas for the energy, Laplacian energy, and signless Laplacian energy of the lintang graph Ln and its line graph L(Ln). An algebraic spectral framework is employed to construct the associated matrices and determine their eigenvalues explicitly. In contrast to general results on line graph energies introduced by Ivan Gutman and subsequent studies, this work focuses on a specific two-hub graph structure for which explicit spectral formulas have not been previously reported. The results show that E(Ln) = 2 sqrt(2n), LE(L1) = QE(L1) = 10 sqrt(3), LE(Ln) = QE(Ln) = 4(n^2 - n + 2)/(n + 2) for n >= 2, and E(L(Ln)) = LE(L(Ln)) = QE(L(Ln)) = 4n - 4. These findings reveal a transition from sublinear to linear growth under the line graph transformation. The results contribute to spectral graph theory and are relevant to chemical graph theory and network analysis.

References

M. A. A. De Freitas, A. S. Bonifácio, M. Robbiano, and B. San Martín, “On matrices associated to directed graphs and applications,” Linear algebra and its applications, vol. 442, pp. 156–164, Feb. 2014, doi: 10.1016/j.laa.2013.07.005.

Q. Li and L. Tang, “Infinite numbers of infinite classes l-borderenergetic graphs,” Match - Communications in mathematical and in computer chemistry, vol. 90, no. 3, pp. 729–742, Jul. 2023, doi: 10.46793/match.90-3.729L.

K. R. Saoub, Graph theory: an introduction to proofs, algorithms, and applications, 1st ed. in Textbooks in mathematics. Boca Raton: CRC Press, 2021. doi: 10.1201/9781138361416.

V. Zverovich, Modern Applications of Graph Theory, 1st ed. Oxford: Oxford university press, 2021. doi: 10.1093/oso/9780198856740.001.0001.

A. Alexander, F. Fran, and N. Kusumastuti, “Complexity of Graphs with Wheel Graph and Fan Graph as their Blocks,” JTAM (Jurnal teori dan aplikasi matematika), vol. 9, no. 4, p. Art. no. 1218, Oct. 2025, doi: 10.31764/jtam.v9i4.32776.

F. Fran, Alexander, Yundari, P. Romanda, and E. Febyolga, “The Complexity of Octopus Graph, Friendship Graph, and Snail Graph,” EduMatSains : Jurnal pendidikan, matematika dan sains, vol. 9, no. 1, pp. 84–101, Jul. 2024, doi: 10.33541/edumatsains.v9i1.6042.

C. Hanssen, F. Fran, and Y. Yundari, “The Complexity of Pencil Graph and Line Pencil Graph,” PYTHAGORAS Jurnal pendidikan matematika, vol. 19, no. 2, pp. 115–125, Nov. 2024, doi: 10.21831/pythagoras.v19i2.77747.

M. R. Zeen El Deen, W. A. Aboamer, and H. M. El-Sherbiny, “The Complexity of the Super Subdivision of Cycle-Related Graphs Using Block Matrices,” Computation, vol. 11, no. 8, p. 162, Aug. 2023, doi: 10.3390/computation11080162.

P. Kornelia, F. Fran, and E. Noviani, “Bilangan B-Kromatik Pada Graf Origami, Graf Lintang, Dan Graf Tadpole,” Bimaster : Buletin ilmiah matematika, statistika dan terapannya, vol. 8, no. 4, Oct. 2019, doi: 10.26418/bbimst.v8i4.36551.

I. Gutman, M. Robbiano, E. A. Martins, D. M. Cardoso, L. Medina, and O. Rojo, “Energy of line graphs,” Linear algebra and its applications, vol. 433, no. 7, pp. 1312–1323, Dec. 2010, doi: 10.1016/j.laa.2010.05.009.

K. Ch. Das and S. A. Mojallal, “Relation between signless Laplacian energy, energy of graph and its line graph,” Linear algebra and its applications, vol. 493, pp. 91–107, Mar. 2016, doi: 10.1016/j.laa.2015.12.006.

H. A. Ganie, B. A. Chat, and S. Pirzada, “Signless Laplacian energy of a graph and energy of a line graph,” Linear algebra and its applications, vol. 544, pp. 306–324, May 2018, doi: 10.1016/j.laa.2018.01.021.

I. Gutman and B. Zhou, “Laplacian energy of a graph,” Linear algebra and its applications, vol. 414, no. 1, pp. 29–37, Apr. 2006, doi: 10.1016/j.laa.2005.09.008.

I. Gutman and B. Furtula, “Survey of Graph Energies,” Mathematics interdisciplinary research, vol. 2, no. 2, Sep. 2017, doi: 10.22052/mir.2017.81507.1057.

A. Raza, M. Munir, M. Hussain, and F. Tasgera, “A Spectrum-Based Approach to Network Analysis Utilizing Laplacian and Signless Laplacian Spectra to Torus Networks,” IEEE Access, vol. 12, pp. 52016–52029, 2024, doi: 10.1109/ACCESS.2024.3384300.

S. Srinivasan and V. Shanmugavel, “Comparative study on different types of energies,” Indian journal of science and technology, vol. 17, no. 28, pp. 2954–2959, Jul. 2024, doi: 10.17485/IJST/v17i28.1447.

Y. Chen, S. Liu, and Z. Wang, “Signless Laplacian energy and spectral radius of a graph,” Discrete applied mathematics, vol. 378, pp. 216–225, Jan. 2026, doi: 10.1016/j.dam.2025.07.015.

P. Bora and M. Rahman, “On The Spectral Characteristics Of Signless Laplacian Matrix,” South east asian journal of mathematics and mathematical sciences, vol. 21, no. 2, pp. 43–52, Aug. 2025, doi: 10.56827/SEAJMMS.2025.2102.3.

G. H. FathTabar, “On the Signless Laplacian Eigenvalues and Optimum SLE of Graph,” Fuzzy optimization and modeling journal, vol. 3, no. 2, pp. 40–45, Jun. 2022, doi: 10.30495/fomj.2022.1950857.1059.

S. Banerjee, “The Signless Laplacian Spread of Power Graphs of Finite Groups,” Ikonion journal of mathematics, vol. 7, no. 2, pp. 24–35, Dec. 2025, doi: 10.54286/ikjm.1658465.

A. Alhevaz, M. Baghipur, S. Pirzada, and Y. Shang, “Some bounds for distance signless Laplacian energy-like invariant of networks,” Carpathian mathematical publications, vol. 17, no. 1, pp. 255–276, Jun. 2025, doi: 10.15330/cmp.17.1.255-276.

U. Preetha P, M. Suresh, and E. Bonyah, “On the spectrum, energy and Laplacian energy of graphs with self-loops,” Heliyon, vol. 9, no. 7, p. e17001, Jul. 2023, doi: 10.1016/j.heliyon.2023.e17001.

A. Bilal and M. Mobeen Munir, “Modified Sombor Spectral Radii and Modified Sombor Energies of Splitting and Shadow Graphs,” Scientific annals of computer science, vol. 34, no. 2, pp. 89–112, 2024, doi: 10.47743/SACS.2024.2.89.

A. Mohamed Atheeque and S. Sharief Basha, “Signless Laplacian energy aware decision making for electric car batteries based on intuitionistic fuzzy graphs,” Science progress, vol. 107, no. 4, p. 00368504241301813, Oct. 2024, doi: 10.1177/00368504241301813.

R. Sarathy and J. R. Sankar, “Laplacian energy and color based energy for graphs associated with commutative rings,” Journal of discrete mathematical sciences and cryptography, vol. 27, no. 8, pp. 2431–2445, 2024, doi: 10.1080/09720529.2024.2338813.

B. A. Rather, M. Aijaz, F. Ali, N. Mlaiki, and A. Ullah, “On distance signless Laplacian eigenvalues of zero divisor graph of commutative rings,” AIMS Mathematics, vol. 7, no. 7, pp. 12635–12649, 2022, doi: 10.3934/math.2022699.

K. Sato and M. Suzuki, “The nearest graph Laplacian in Frobenius norm,” Numerical linear algebra with applications, vol. 31, no. 6, p. e2578, Dec. 2024, doi: 10.1002/nla.2578.

N. A. Mohd Rizal, N. I. Alimon, M. Selvarajoo, and N. H. Sarmin, “Laplacian Spectrum and Laplacian Energy of the Zero Divisor Graph for Z_3alpha,” Malaysian journal of fundamental and applied sciences, vol. 21, no. 6, pp. 2926–2936, Dec. 2025, doi: 10.11113/mjfas.v21n6.4303.

N. F. Yalçın, “On Laplacian Energy of r-Uniform Hypergraphs,” Symmetry, vol. 15, no. 2, p. 382, Feb. 2023, doi: 10.3390/sym15020382.

N. Nurliantika, F. Fran, and Y. Yundari, “Determinants of Tridiagonal and Circulant Matrices Special Form by Chebyshev Polynomials,” JTAM (Jurnal teori dan aplikasi matematika), vol. 9, no. 1, p. 190, Jan. 2025, doi: 10.31764/jtam.v9i1.27871.

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Published

2026-07-29

Issue

Vol. 10 No. 2 (2026): Zero: Jurnal Sains Matematika dan Terapan

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