The Laplacians, Kirchhoff index and complexity of linear M\"obius and cylinder octagonal-quadrilateral networks

TL;DR

This paper derives explicit formulas for Kirchhoff indices and complexity of linear octagonal-quadrilateral networks and their M"obius and cylinder variants, revealing their spectral properties and asymptotic relationships.

Contribution

It provides new explicit formulas for Kirchhoff indices and complexity of specific network topologies using Laplacian spectra, expanding spectral graph theory applications.

Findings

Kirchhoff index of M"obius network is about one-third of its Wiener index as n approaches infinity.

Explicit formulas for Kirchhoff indices and complexity are derived using Laplacian characteristic polynomials.

The spectral properties reveal asymptotic relationships between different network invariants.

Abstract

Spectrum graph theory not only facilitate comprehensively reflect the topological structure and dynamic characteristics of networks, but also offer significant and noteworthy applications in theoretical chemistry, network science and other fields. Let $L_{n}^{8,4}$ represent a linear octagonal-quadrilateral network, consisting of $n$ eight-member ring and $n$ four-member ring. The M\"{o}bius graph $Q_{n}(8,4)$ is constructed by reverse identifying the opposite edges, whereas cylinder graph $Q'_{n}(8,4)$ identifies the opposite edges by order. In this paper, the explicit formulas of Kirchhoff indices and complexity of $Q_{n}(8,4)$ and $Q'_{n}(8,4)$ are demonstrated by Laplacian characteristic polynomials according to decomposition theorem and Vieta's theorem. In surprise, the Kirchhoff index of $Q_{n}(8,4)$($Q'_{n}(8,4)$) is approximately one-third half of its Wiener index as $n\to\infty$.