On the Harmonic characteristic polynomial of specific graphs

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.