Properties of Laplacian Eigenvalues of Some Bicyclic and Tricyclic Graphs

Document Type : Research Paper

Authors

Department of Mathematics‎, ‎Abbottabad University of Science \& Technology‎, ‎Havelian‎, ‎KPK‎, ‎Pakistan

10.22052/ijmc.2025.256371.1984

Abstract

‎The Laplacian energy (LE) and the Laplacian energy-like (LEL) have recently been proposed based on molecular graph analogues of the total $\pi$-electron energy E‎. ‎Both energies have been widely studied recently because of their wide range of applications‎. ‎In the present work‎, ‎exact expressions of the Laplacian energy and the Laplacian-like invariants of bicyclic and tricyclic molecular graphs in terms of their orders have been obtained‎. ‎We also compute these expressions for the complements of these classes of graphs‎. ‎It is shown that LEL is strictly less than LE for these classes of molecular graphs‎, ‎but for their complements the inequality is the opposite‎.

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References

[1] I. Gutman, Total Pi-Electron energy of Benzenoid Hydrocarbons, Topics Curr. Chem. 162 (1992) 29–63.
[2] I. Gutman, Hyper-Wiener index and Laplacian spectrum, J. Serb. Chem. Soc. 68 (2003) 949–952.
[3] I. Gutman, B. Furtula, J. Ðurdevic, R. Kovacevic and S. Stankovic, Annelated perylenes: benzenoid molecules violating the Kekule-structure-based cyclic conjugation models, J. Serb. Chem. Soc. 70 (2005) 1023–1032.
[4] M. Peric, I. Gutman and J. Radic-Peric, The Huckel total pielectron energy puzzle, J. Serb. Chem. Soc. 71 (2006) 771–783, https://doi.org/10.2298/JSC0607771P.
[5] A. Graovac, I. Gutman and N. Trinajstic, Topological Approach to the Chemistry of Conjugated Molecules, Springer-Verlag, Berlin, 1977.
[6] I. Gutman and O. E. Polansky, Mathematical Concept in Organic Chemistry, Springer-Verlag, Berlin, 1986.
[7] I. Gutman, N. Trinajstic and C. F. Wilcox, Graph theory and molecularorbitals .XII. Acyclic Polyenes, J. Chem. Phys. 62 (1975) 3399–3405.
[8] W. Xiao and I. Gutman, Resistance distance and Laplacian spectrum, Theor. Chem. Acc. 110 (2003) 284–289, https://doi.org/10.1007/s00214-003-0460-4.
[9] B. Mohar, Laplacian matrices of graphs, Stud. Phys. Theor. Chem. 63 (1989) 1–8.
[10] I. Gutman and B. Zhou, Laplacian energy of a graph, Linear Algebra Appl. 414 (2006) 29–37, https://doi.org/10.1016/j.laa.2005.09.008.
[11] B. Zhou and I. Gutman, On Laplacian energy of graphs, MATCH Commun. Math. Comput. Chem. 57 (2007) 211–220.
[12] E. Milovanovic, I. Milovanovic and M. Matejic, On relation between the Kirchhoff index and Laplacian-Energy-Like invariant of graphs, Math. Interdisc. Res. 2 (2017) 141–154, https://doi.org/10.22052/MIR.2017.85687.1063.
[13] N. De, On eccentricity version of Laplacian energy of a graph, Math. Interdisc. Res. 2 (2017) 131–139, https://doi.org/10.22052/MIR.2017.70534.1051.
[14] J. Liu and B. Liu, A Laplacian-energy-like invariant of a graph, MATCH Commun. Math. Comput. Chem. 59 (2008) 355 372.
[15] W. Wang and Y. Luo, On Laplacian-energy-like invariant of a graph, Linear Algebra Appl. 437 (2012) 713–721, https://doi.org/10.1016/j.laa.2012.03.004.
[16] B. X. Zhu, The Laplacian-energy like of Graphs, Appl. Math. Lett. 24 (2011) 1604–1607, https://doi.org/10.1016/j.aml.2011.04.010.
[17] I. Gutman, Some properties of Laplacian eigenvectors , Bull. Cl. Sci. Math. Nat. Sci. Math. (2003) 1–6.
[18] S. Radenkovic and I. Gutman, Total -electron energy and Laplacian energy: How far the analogy goes?, J. Serb. Chem. Soc. 72 (2007) 1343–1350.
Iranian Journal of Mathematical Chemistry
Volume 17, Issue 1
In Progress
March 2026
Pages 13-33

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