Computing the Laplacian spectrum and Wiener index of pentagonal-derivation cylinder/Möbius network

Umar Ali; Junxiang Li; Yasir Ahmad; Zahid Raza

doi:10.1016/j.heliyon.2024.e24182

Computing the Laplacian spectrum and Wiener index of pentagonal-derivation cylinder/Möbius network

Heliyon. 2024 Jan 10;10(2):e24182. doi: 10.1016/j.heliyon.2024.e24182. eCollection 2024 Jan 30.

Authors

Umar Ali¹, Junxiang Li¹, Yasir Ahmad², Zahid Raza³

Affiliations

¹ Business School, University of Shanghai for Science and Technology, Shanghai, 200093, China.
² School of Mathematical Sciences, Anhui University, Hefei, Anhui, 230601, China.
³ Department of Mathematics, College of Sciences, University of Sharjah, United Arab Emirates.

Abstract

The Laplacian spectrum significantly contributes the study of the structural features of non-regular networks. Actually, it emphasizes the interaction among the network eigenvalues and their structural properties. Let $P_{n}$ $(P_{n}^{'})$ represent the pentagonal-derivation cylinder (Möbius) network. In this article, based on the decomposition techniques of the Laplacian characteristic polynomial, we initially determine that the Laplacian spectra of $P_{n}$ contain the eigenvalues of matrices $L_{R}$ and $L_{S}$ . Furthermore, using the relationship among the coefficients and roots of these two matrices, explicit calculations of the Kirchhoff index and spanning trees of $P_{n}$ are determined. The relationship between the Wiener and Kirchhoff indices of $P_{n}$ is also established.

Keywords: 05C12; 05C50; 15A09; Kirchhoff index; Pentagonal-derivation cylinder/Möbius; Spanning tree; Wiener index.