On the Harmonic characteristic polynomial of specific graphs

Sadruddin Rahimi; Saeid Alikhani

paper

On the Harmonic characteristic polynomial of specific graphs

arXiv:2511.12734

Abstract

This paper explores the Harmonic matrix $M H (G)$ 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 $H E (G)$ of a graph as the sum of the absolute values of the eigenvalues of $M H (G)$ . Explicit expressions for both the Harmonic characteristic polynomial and the Harmonic energy are derived for several specific classes of graphs.

16 pages, 4 figures