The Laplacian spectrum, Kirchhoff index and complexity of the linear heptagonal networks

TL;DR

This paper analyzes the eigenvalues of linear heptagonal networks to derive explicit formulas for their Kirchhoff index and total complexity, linking spectral properties to structural network measures.

Contribution

It provides a novel spectral analysis approach to compute Kirchhoff index and complexity for linear heptagonal networks using Laplacian polynomial decomposition.

Findings

Explicit formulas for Kirchhoff index of $H_n$

Explicit formulas for total complexity of $H_n$

Eigenvalues derived from matrices $L_A$ and $L_S$

Abstract

Let $H_n$ be the linear heptagonal networks with $2n$ heptagons. We study the structure properties and the eigenvalues of the linear heptagonal networks. According to the Laplacian polynomial of $H_n$, we utilize the decomposition theorem. Thus, the Laplacian spectrum of $H_n$ is created by eigenvalues of a pair of matrices: $L_A$ and $L_S$ of order number $5n+1$ and $4n+1$, respectively. On the basis of the roots and coefficients of their characteristic polynomials of $L_A$ and $L_S$, we not only get the explicit forms of Kirchhoff index, but also corresponding total complexity of $H_n$.