Harmonic Matrix MH(G): Spectral Invariants

• Harmonic Matrix MH(G) is a symmetric matrix defined for a graph where adjacent vertices yield entries 2/(d_i+d_j), capturing a degree-weighted bond between nodes.
• It leads to the harmonic characteristic polynomial and harmonic energy, unifying classical invariants such as the Fajtlowicz harmonic index and Gutman’s graph energy.
• Explicit formulas for families like complete, star, and bipartite graphs, along with open research challenges, underscore its significance in spectral graph theory.

The harmonic matrix $MH(G)$ is a graph-theoretic construct arising from the paper of spectral invariants that unify the Fajtlowicz harmonic index and Gutman’s graph energy. For a simple undirected graph $G$ with degree sequence $\{d_i\}$, $MH(G)$ is the symmetric $n \times n$ real matrix where each entry is $2/(d_i + d_j)$ if vertices $v_i$ and $v_j$ are adjacent, and zero otherwise. This matrix facilitates a spectral analysis distinct from the adjacency or Laplacian formalisms. Key invariants derived from $MH(G)$ include the harmonic characteristic polynomial and the harmonic energy $HE(G)$, defined as the sum of the absolute values of its eigenvalues. These invariants admit explicit formulas for several notable graph families and are the subject of ongoing research regarding their extremal properties and relationships to graph structure (Rahimi et al., 16 Nov 2025).

1. Definition and Construction

Let $G=(V,E)$ be a simple undirected graph of order $n$, with vertex set $V = \{v_1,\dots,v_n\}$ and degrees $d_i = \deg(v_i)$. The harmonic matrix is defined as

$$\mathrm{MH}(G) = (h_{ij})_{1 \leq i,j \leq n},\quad h_{ij} = \begin{cases} \displaystyle \frac{2}{d_i + d_j}& \text{if } v_i \sim v_j,\ 0 & \text{otherwise}. \end{cases}$$

Equivalently, in terms of the indicator function $\chi_{\{v_i\sim v_j\}}$,

$$\mathrm{MH}(G)_{i,j} = \frac{2}{d_i+d_j}\,\chi_{\{v_i\sim v_j\}}.$$

The matrix is real, symmetric, and has sparsity reflecting the adjacency structure of $G$, with edge weights determined by local degree sums.

2. Harmonic Characteristic Polynomial and Eigenvalues

The harmonic characteristic polynomial is

$$\Phi_{\mathrm{MH}(G)}(\lambda) = \det(\lambda I_n - \mathrm{MH}(G)) = \prod_{i=1}^n (\lambda - \gamma_i),$$

where $\gamma_1 \geq \cdots \geq \gamma_n$ are the real eigenvalues of $MH(G)$. No general closed-form for arbitrary graphs exists, but for many root graph families explicit factorizations are obtainable via block decompositions or determinant recursions on tridiagonal blocks (Rahimi et al., 16 Nov 2025).

Notably, for $G' \subset G$ induced, the eigenvalues of $MH(G')$ interlace those of $MH(G)$. General spectral graph theory bounds apply: the spectrum of $MH(G)$ lies in $[-1,1]$ due to the boundedness of off-diagonal entries and row sums.

3. Explicit Results for Standard Graph Families

For the following canonical graph families, exact forms of the harmonic characteristic polynomial, eigenvalues, and corresponding harmonic energies are established:

Graph Family	$\Phi_{MH(G)}(\lambda)$	$\text{Spec}$ or $HE(G)$
Path $P_n$ ($n\geq5$)	$$\lambda\Lambda_{n-2}-\frac{8}{9}\lambda\Lambda_{n-3}+\frac{16}{81}\Lambda_{n-4}$$	$HE(P_n)$ from real roots; no closed form
Cycle $C_n$ ($n\geq3$)	$$\lambda\Lambda_{n-1}-\tfrac{1}{2}\Lambda_{n-2}-(\frac{1}{2})^{n-1}$$	$HE(C_n)$ from block analysis
Star $S_n$	$\lambda^{n-2}(\lambda^2-\frac{4(n-1)}{n^2})$	$\{0^{(n-2)},\pm\frac{2\sqrt{n-1}}{n}\}$; $HE(S_n)=\frac{4\sqrt{n-1}}{n}$
Complete $K_n$	$(\lambda-1)(\lambda+\frac{1}{n-1})^{n-1}$	$\{1,-\frac{1}{n-1}^{(n-1)}\}$; $HE(K_n)=2$
Complete bipartite $K_{m,n}$	$\lambda^{m+n-2}(\lambda^2-\frac{4mn}{(m+n)^2})$	$\{0^{(m+n-2)},\pm\frac{2\sqrt{mn}}{m+n}\}$; $HE(K_{m,n})=\frac{4\sqrt{mn}}{m+n}$
Friendship $F_n$	$$(\lambda-\frac{1}{2})^{n-1}(\lambda+\frac{1}{2})^n[\lambda^2-\frac{1}{2}\lambda-\frac{2n}{(n+1)^2}]$$	$HE(F_n) = n$
Dutch windmill $D_m^n$	$\Lambda_{m-1}^{n-1}\Phi_{MH(C_m)}(\lambda)$	See explicit cases $m=4,5$
Book $B_n$	$(\lambda^2-\frac{1}{2})^{n-1}$ times quadratic factors	$HE(B_n)=\frac{n^2+n+2}{n+1}$

Here $\Lambda_k$ are defined recursively for tridiagonal matrices (see Section 4.1 of (Rahimi et al., 16 Nov 2025)).

For star, complete, and complete bipartite graphs, the harmonic spectrum is highly degenerate. The block determinant methods and Schur complement arguments enable the explicit calculations. For path and cycle graphs, recursions on tridiagonal determinants yield the closed form.

4. Harmonic Energy: Definition and Analytical Results

The harmonic energy is

$HE(G) = \sum_{i=1}^n |\gamma_i|,$

where $\gamma_1,\dots,\gamma_n$ are the eigenvalues of $MH(G)$. Closed-form expressions for $HE(G)$ are obtained for the families above. For example:

• $HE(K_n) = 2$
• $HE(S_n) = 4 \sqrt{n-1} / n$
• $HE(K_{m,n}) = 4\sqrt{mn}/(m+n)$
• $HE(F_n)=n$
• $HE(B_n)=(n^2+n+2)/(n+1)$

In all cases, $0 \leq HE(G) \leq n$. For regular graphs, interlacing bounds on $HE(G)$ apply (the case $d=3$ and order $10$ is detailed in Section 4 of (Rahimi et al., 16 Nov 2025)). The harmonic energy captures both global adjacency and degree structure, unifying two classical invariants: the Fajtlowicz harmonic index and Gutman’s graph energy.

5. Fully Worked Example: Complete Graph $K_4$

For $K_4$, every vertex has degree $3$: $MH(K_4) = \frac{1}{3}(J_4 - I_4),$ where $J_4$ is the $4\times4$ all-ones matrix. Thus,

$$\lambda I_4 - MH(K_4) = \left(\lambda + \frac{1}{3}\right)I_4 - \frac{1}{3}J_4.$$

Since $\text{Spec}(J_4) = \{4,0,0,0\}$,

$$\Phi_{MH(K_4)}(\lambda) = (\lambda-1)(\lambda+\tfrac{1}{3})^3,$$

yielding eigenvalues $1$ and $-\frac{1}{3}$ (algebraic multiplicity $3$). The harmonic energy is

$HE(K_4) = |1| + 3\left|-\tfrac{1}{3}\right| = 2.$

6. Open Problems and Research Directions

The investigation of $MH(G)$ and its invariants raises several open questions:

• Extension of the spectral and energy analysis to arbitrary $d$-regular graphs (beyond $d=3$).
• Determination of extremal graphs maximizing or minimizing $HE(G)$ under constraints such as fixed order, size, or degree sequence.
• Development of interlacing and majorization techniques for general spectral bounds of $MH(G)$.
• Characterization of $HE$-uniqueness: the identification of graph classes for which harmonic energy is a complete isomorphism invariant (e.g., for trees, bipartite graphs, cographs).

The harmonic matrix framework provides a unified spectral perspective on previously disparate graph invariants, suggesting rich interplay between degree-weighted adjacency and energy-type quantities and prompting further research on extremal and structural graph theory questions (Rahimi et al., 16 Nov 2025).