Topological indices of linear crossed phenylenes with respect to their Laplacian and normalized Laplacian spectrum

Zhi-Yu Shi; Jia-Bao Liu; Zhi-Yu Shi; Jia-Bao Liu

doi:10.3934/math.2024262

AIMS Mathematics

2024, Volume 9, Issue 3: 5431-5450. doi: 10.3934/math.2024262

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Research article

Topological indices of linear crossed phenylenes with respect to their Laplacian and normalized Laplacian spectrum

Zhi-Yu Shi ^{1
,
,},
Jia-Bao Liu ²

1.
Engineering Department, Anhui Sanlian University, Hefei 230601, China
2.
School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China

Received: 23 November 2023 Revised: 31 December 2023 Accepted: 10 January 2024 Published: 29 January 2024
MSC : 05C50, 05C90

As a powerful tool for describing and studying the properties of networks, the graph spectrum analyses and calculations have attracted substantial attention from the scientific community. Let $ C_{n} $ represent linear crossed phenylenes. Based on the Laplacian (normalized Laplacian, resp.) polynomial of $ C_{n} $, we first investigated the Laplacian (normalized Laplacian, resp) spectrum of $ C_{n} $ in this paper. Furthermore, the Kirchhoff index, multiplicative degree-Kirchhoff, index and complexity of $ C_{n} $ were obtained through the relationship between the roots and the coefficients of the characteristic polynomials. Finally, it was found that the Kirchhoff index and multiplicative degree-Kirchhoff index of $ C_{n} $ were approximately one quarter of their Wiener index and Gutman index, respectively.
- Laplacian spectrum,
- normalized Laplacian spectrum,
- Kirchhoff index,
- multiplicative degree-Kirchhoff index,
- complexity
Citation: Zhi-Yu Shi, Jia-Bao Liu. Topological indices of linear crossed phenylenes with respect to their Laplacian and normalized Laplacian spectrum[J]. AIMS Mathematics, 2024, 9(3): 5431-5450. doi: 10.3934/math.2024262

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Abstract

As a powerful tool for describing and studying the properties of networks, the graph spectrum analyses and calculations have attracted substantial attention from the scientific community. Let $ C_{n} $ represent linear crossed phenylenes. Based on the Laplacian (normalized Laplacian, resp.) polynomial of $ C_{n} $, we first investigated the Laplacian (normalized Laplacian, resp) spectrum of $ C_{n} $ in this paper. Furthermore, the Kirchhoff index, multiplicative degree-Kirchhoff, index and complexity of $ C_{n} $ were obtained through the relationship between the roots and the coefficients of the characteristic polynomials. Finally, it was found that the Kirchhoff index and multiplicative degree-Kirchhoff index of $ C_{n} $ were approximately one quarter of their Wiener index and Gutman index, respectively.

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