Spectral Sum of Trees

• Spectral Sum of Trees is defined as the sum of the two largest eigenvalues of a tree’s adjacency matrix, elucidating its key structural properties.
• Extremal results identify balanced double comets as maximizers and paths or stars as minimizers depending on the vertex count.
• The analysis extends to spectral moments and functionals, linking closed walks to practical applications in chemistry, physics, and combinatorics.

The spectral sum of trees concerns fundamental questions about the extremal properties of adjacency matrix eigenvalues for trees. Let $T$ be a tree with $n$ vertices and adjacency matrix $A(T)$, with spectrum $$\lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_n$$. The spectral sum $S(T)$ is defined as $S(T) = \lambda_1 + \lambda_2$. This concept generalizes to spectral moments and other spectral functionals, underpinning several classical problems in algebraic graph theory with applications in chemistry, physics, and combinatorics.

1. Definitions and Core Notation

Given a simple undirected tree $T$ on $n$ vertices, the adjacency matrix $A(T)$ has real eigenvalues $\lambda_1,\dots,\lambda_n$. The spectral sum is

$S(T) = \lambda_1 + \lambda_2,$

where the eigenvalues are ordered nonincreasingly. The $k$th spectral moment is

$M_k(T) = \sum_{i=1}^n \lambda_i^k,$

coinciding with the trace $\mathrm{tr}(A^k)$ and, combinatorially, with the number of closed walks of length $k$ in $T$ (Andriantiana et al., 2013).

2. Extremal Spectral Sums Among Trees

2.1 Maximum of $S(T)$

Kumar, Mohar, Pragada, and Zhan established that for every $n \geq 5$, the maximum $S(T)$ is achieved uniquely by the balanced double comet

$\operatorname{DC}(k_1, k_2, 3)$

with $$k_1 = \lfloor (n-3)/2 \rfloor, k_2 = \lceil (n-3)/2 \rceil$$, constructed by a path of length 3 and attaching $k_1$ leaves at one end and $k_2$ at the other. The two largest eigenvalues are the positive square roots of the largest two roots of the quartic equation

$x^4 - (n-1)x^2 + (k_1 k_2 + k_1 + k_2) = 0.$

Closed formulas exist for both even and odd $n$, yielding

• For $n = 2m+1$: $\lambda_1 = \sqrt{(n+1)/2}$, $\lambda_2 = \sqrt{(n-3)/2}$
• For $n = 2m$: $\lambda_1 = \sqrt{\frac{1}{2}(n-1 + \sqrt{5})}$, $\lambda_2 = \sqrt{\frac{1}{2}(n-1 - \sqrt{5})}$ (Kumar et al., 15 Jan 2026).

2.2 Minimum of $S(T)$

For every $n \geq 16$, the minimum spectral sum among $n$-vertex trees is attained uniquely by the path $P_n$, whose spectral sum admits a closed form: $$\lambda_1 = 2\cos\left(\frac{\pi}{n+1}\right), \quad \lambda_2 = 2\cos\left(\frac{2\pi}{n+1}\right), \quad S(P_n) = 2[\cos(\frac{\pi}{n+1}) + \cos(\frac{2\pi}{n+1})].$$ For $n \leq 15$, the star $K_{1,n-1}$ achieves the minimum, with $S(K_{1,n-1}) = \sqrt{n-1}$ (Kumar et al., 15 Jan 2026).

3. Spectral Sums, Moments, and Majorization

Spectral moments,

$M_k(T) = \sum_{i=1}^n \lambda_i^k,$

encode structural information about closed walks in $T$. Given any prescribed degree sequence $D$, the “greedy tree” $G(D)$ is constructed via breadth-first attachment, always placing the largest available degrees closest to the root. The greedy tree is extremal: $$M_k(T) \leq M_k(G(D)) \quad \forall T \in \mathcal{T}_D,\;\forall k\geq 0.$$ This is sharpened by majorization: if $B \preccurlyeq D$, then $M_k(G(B)) \leq M_k(G(D))$ for all $k$. The proof relies on induction over walk structures and “branch-swapping” arguments (Andriantiana et al., 2013).

These results extend to analytic spectral sums $E_f(T) = \sum_{i=1}^n f(\lambda_i)$ where the even Taylor coefficients $a_k$ of $f$ are nonnegative. As corollaries, the greedy tree maximizes invariants such as the Estrada index

$$EE(T) = \sum_{i=1}^n e^{\lambda_i} = \sum_{k=0}^\infty \frac{M_k(T)}{k!}$$

and the graph energy $\sum |\lambda_i|$.

4. Convex Combination of Leading Eigenvalues

For $\alpha \in [0,1]$, the functional

$$\Psi(T, \alpha) = \alpha\lambda_1(T) + (1-\alpha)\lambda_2(T)$$

is maximized, over all $n$-vertex trees, by a double comet whose parameters interpolate between balanced and unbalanced forms as $\alpha$ varies. Precisely,

• For $0 \leq \alpha \leq \tfrac{1}{2}$, the balanced double comet $DC(k,k,3)$ or variants remain extremal.
• For $\tfrac{1}{2} < \alpha < 1$, the extremal double comet moves leaves asymmetrically between branches.
• For $\alpha \to 1$, the star $K_{1,n-1}$ is the unique maximizer.

Asymptotic analysis for large $n$ gives: $$\Psi_n(\alpha) / \sqrt{n-1} \to \begin{cases} \sqrt{1/2} & \text{if}~\alpha \leq 1/2,\ \sqrt{\alpha^2 + (1-\alpha)^2} & \text{if}~\alpha \geq 1/2. \end{cases}$$ (Kumar et al., 15 Jan 2026).

5. Small $n$ and Explicit Spectra

For lower orders, the extremal trees shift depending on $n$:

$n$	$S(K_{1,n-1})$	$S(P_n)$	Maximizer	Minimizer
4	$\sqrt{3}$	2[cos(π/5)+cos(2π/5)] ≈ 2.236	$P_4$	$K_{1,3}$
5	2.000	2[cos(π/6)+cos(2π/6)] = 2.732	$P_5$	$K_{1,4}$
6	≈ 2.236	$\lambda_1 ≈ 1.8019, \lambda_2 ≈ 1.2469$	$DC(1,2,3)$	$K_{1,5}$

The double comet structure becomes strictly extremal for $n \geq 5$, and the path is minimal for $n \geq 16$ (Kumar et al., 15 Jan 2026).

6. Applications and Generalizations

Spectral sums and moments quantify closed walks and relate to diverse invariants. In chemistry, these connect to indices such as the Estrada index (protein folding) and graph energy (molecular stability) (Andriantiana et al., 2013). Extremal results for spectral sums identify trees maximizing or minimizing walk-richness or related substructure counts under degree constraints.

For fixed maximum degree $\Delta$, the Volkmann tree—a maximally balanced $\Delta$-ary tree—achieves maximal even spectral moments and Estrada index, confirming conjectures of Ilić–Stevanović and Gutman–Furtula–Marković–Glišić (Andriantiana et al., 2013).

These techniques extend beyond trees with a fixed degree sequence to trees with majorized degree sequences, and may be adapted for broader classes, including unicyclic or bipartite graphs and beyond.

7. Proof Techniques and Theoretical Significance

Extremal spectral sums are established via:

• Inequalities on the sum of squares of leading eigenvalues, combined with the trace bound $\lambda_1^2 + \lambda_2^2 \leq 2(n-1)$ for trees.
• “Kelmans operations” and “rotation” arguments to control and optimize eigenvalue placement in double comets.
• Majorization, induction on walk structures, and branch-swapping to demonstrate maximality of greedy trees in spectral moments.
• Analysis of eigenvector structures, especially of $\lambda_2$, to rule out branching and confirm path minimality.

The unified framework highlights the deep relationship between tree topology, degree sequences, spectral moments, and walk counts, bridging combinatorial, analytic, and algebraic aspects of spectral graph theory (Kumar et al., 15 Jan 2026, Andriantiana et al., 2013).