On the Harmonic characteristic polynomial of specific graphs

TL;DR

This paper studies the spectral properties of the Harmonic matrix of graphs, deriving explicit formulas for its characteristic polynomial and energy for specific graph classes.

Contribution

It introduces the Harmonic matrix and provides new explicit formulas for its characteristic polynomial and energy for certain graphs.

Findings

Explicit formulas for Harmonic characteristic polynomial

Expressions for Harmonic energy of specific graphs

Spectral analysis of the Harmonic matrix

Abstract

This paper explores the Harmonic matrix $MH(G)$ associated with a simple graph $ G $, where each entry corresponds to $ \frac{2}{d_i + d_j} $ for adjacent vertices $ v_i $ and $ v_j $. We investigate the spectral properties of this matrix, particularly focusing on its eigenvalues. A central objective of this work is to compute the Harmonic characteristic polynomial. Furthermore, we analyze the Harmonic energy $ HE(G) $ of a graph as the sum of the absolute values of the eigenvalues of $ MH(G) $. Explicit expressions for both the Harmonic characteristic polynomial and the Harmonic energy are derived for several specific classes of graphs.