On the Harmonic characteristic polynomial of specific graphs
Sadruddin Rahimi, Saeid Alikhani
TL;DR
This work defines the Harmonic matrix $H(G)$ for simple graphs via $r_{ij} = \frac{2}{d_i+d_j}$ on edges and studies its spectral invariants through the Harmonic characteristic polynomial $\phi_H(G,\lambda)$ and the Harmonic energy $HE(G) = \sum_i |\gamma_i|$. It develops determinant-based methods to obtain explicit $\phi_H$ and $HE$ for a broad set of graph families, including stars, complete graphs, complete bipartite graphs, friendship and Dutch Windmill graphs, and book graphs, with key closed forms such as $\phi_H(S_n,\lambda) = \lambda^{n-2}(\lambda^2 - \tfrac{4(n-1)}{n^2})$ and $\phi_H(K_n,\lambda)=(\lambda - 1)(\lambda + \tfrac{1}{n-1})^{n-1}$. For 3-regular graphs, the paper shows $HE$ is additive over components and provides a complete enumeration framework for order-10 cubic graphs, highlighting the Petersen graph as maximizing $HE$ within its class while not being $HE$-unique. Overall, the results extend spectral graph theory by delivering explicit Harmonic polynomials and energies for diverse graph families, enabling finer graph comparisons and classifications by harmonic spectra.
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.