On the Harmonic characteristic polynomial of specific graphs

2025/11/16 by Sadegh Rahimi, Rahimi, Sadruddin, ‎Saeid Alikhani +1
Mathematics · Physics and Astronomy · Computer Science · #Graph theory and applications #Complex Network Analysis Techniques #Graph Labeling and Dimension Problems

paper · pdf · doi:10.48550/arxiv.2511.12734

Abstract

This paper explores the Harmonic matrix MH(G) associated with a simple graph G , where each entry corresponds to (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.

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