Spectral Monotonicity under Leaf Attachment and Limiting Behavior in Discrete Einstein Trees

Abstract: Let $R_T$ be the Ricci matrix of a finite tree $T$ introduced in \cite{BaiChengHua2026}, the largest eigenvalue $λ<em>{\max}(R_T)$ determines the sign of a discrete Einstein metric curvature on the tree. This paper investigates the asymptotic behavior of the sequence $λ_k = λ</em>{\max}(R_{T_k})$ obtained by repeatedly adding pendant edges at a fixed vertex. We prove that $λ<em>k$ converges to a limit $λ</em>\infty$ that depends only on the local branch data of $T$, and establish a first-order asymptotic expansion: [ λk = λ\infty + \fracα{d+k} + O!\left(\frac{1}{(d+k)^2}\right), ] where $d$ is the degree of the original vertex, and the coefficient $α$ is given by a spectral projection. As a corollary, when $α\neq 0$, $λ_k$ is eventually strictly monotonic (increasing or decreasing). This theory reveals the fine influence of local leaf addition on the global spectrum.

• The paper proves that the largest Ricci-matrix eigenvalue converges under repeated leaf attachment to the maximum of zero and the largest Dirichlet eigenvalue of the tree’s branches.
• It reduces the problem to a finite-dimensional perturbation, showing that the first-order coefficient α determines whether eigenvalues eventually increase toward or decrease toward the limit.
• A numerical example demonstrates that convergence need not be globally monotone, with eigenvalues crossing from negative to positive before eventually declining toward zero.

Background and motivation

For a finite tree $T=(V,E)$, the Ricci matrix $R_T\in\mathbb{R}^{E\times E}$ introduced by Bai, Cheng, and Hua encodes the Lin–Lu–Yau optimal-transport curvature of weighted trees: a discrete Einstein metric (edge weights with constant curvature) exists on $T$ if and only if the Perron eigenvector of $R_T$ is positive, and the Einstein curvature equals $\kappa=-\lambda_{\max}(R_T)$ (Bai et al., 24 Apr 2026). The sign of $\lambda_{\max}(R_T)$ therefore determines the sign of the curvature. Prior work established that $\lambda_{\max}(R_T)\le 0$ forces $T$ to be a caterpillar, that attaching a leaf at a vertex of degree at most two does not decrease $\lambda_{\max}$, and that when $\lambda_{\max}(R_T)<0$, leaf attachment at any vertex strictly increases it; Cheng subsequently classified all trees with nonpositive Perron eigenvalue and confirmed that the only infinite zero-curvature family is $R_T\in\mathbb{R}^{E\times E}$0 (Cheng, 20 May 2026). These results motivate the question addressed here: under repeated attachment of pendant edges at a fixed vertex $R_T\in\mathbb{R}^{E\times E}$1, does $R_T\in\mathbb{R}^{E\times E}$2 converge, and is it eventually monotone?

The paper answers both questions affirmatively in a precise sense. The main structural observation is that the Perron eigenvector of $R_T\in\mathbb{R}^{E\times E}$3 must be constant on the $R_T\in\mathbb{R}^{E\times E}$4 new pendant edges (they form an orbit under automorphisms commuting with $R_T\in\mathbb{R}^{E\times E}$5), so $R_T\in\mathbb{R}^{E\times E}$6 reduces to the largest eigenvalue of a fixed-dimensional matrix $R_T\in\mathbb{R}^{E\times E}$7 admitting the affine decomposition

$R_T\in\mathbb{R}^{E\times E}$8

where $R_T\in\mathbb{R}^{E\times E}$9 and $T$0 is independent of $T$1. All subsequent results flow from this representation.

One-step leaf addition: Rayleigh difference formula

The paper first develops a variational analysis of a single leaf addition using the vertex decomposition of the quadratic form,

$T$2

with $T$3 and $T$4. Extending $T$5 by a value $T$6 on the new edge yields the exact difference formula

$T$7

Maximizing over $T$8 produces the $T$9-sharp criterion: for a unit vector $R_T$0 with $R_T$1 and $R_T$2, if

$R_T$3

then some extension attains Rayleigh quotient at least $R_T$4, whence $R_T$5. Applying this to the Perron vector gives a sufficient condition for one-step monotonicity. Since $R_T$6, the worst case $R_T$7 yields a degree-only threshold $R_T$8 for $R_T$9 (and $\kappa=-\lambda_{\max}(R_T)$0 for $\kappa=-\lambda_{\max}(R_T)$1): whenever $\kappa=-\lambda_{\max}(R_T)$2, attaching a leaf at $\kappa=-\lambda_{\max}(R_T)$3 cannot decrease the Perron eigenvalue. This recovers coarse degree-dependent information while retaining the sharper $\kappa=-\lambda_{\max}(R_T)$4-based test.

The limit formula via Dirichlet decoupling

Removing $\kappa=-\lambda_{\max}(R_T)$5 splits $\kappa=-\lambda_{\max}(R_T)$6 into branches $\kappa=-\lambda_{\max}(R_T)$7. On the orbit-reduced subspace, each diagonal block $\kappa=-\lambda_{\max}(R_T)$8 of $\kappa=-\lambda_{\max}(R_T)$9 is obtained from $\lambda_{\max}(R_T)$0 by setting $\lambda_{\max}(R_T)$1 to zero—equivalently, imposing a Dirichlet boundary condition at $\lambda_{\max}(R_T)$2 or taking $\lambda_{\max}(R_T)$3. In particular, the root-edge diagonal entry becomes $\lambda_{\max}(R_T)$4 rather than $\lambda_{\max}(R_T)$5. The interface vectors satisfy $\lambda_{\max}(R_T)$6 (supported only on the root edge), $\lambda_{\max}(R_T)$7, and $\lambda_{\max}(R_T)$8.

The resulting limit matrix $\lambda_{\max}(R_T)$9 is block upper-triangular with diagonal blocks $\lambda_{\max}(R_T)\le 0$0 and a scalar zero block, so its spectrum is $\lambda_{\max}(R_T)\le 0$1, giving the limit formula

$\lambda_{\max}(R_T)\le 0$2

Two consequences deserve emphasis. First, the growing leaf cluster contributes a floor value of exactly $\lambda_{\max}(R_T)\le 0$3: since the symmetric mode of the new leaves has Rayleigh quotient $\lambda_{\max}(R_T)\le 0$4, repeated leaf addition can never drive the limit negative even if every branch is subcritical ($\lambda_{\max}(R_T)\le 0$5). Second, $\lambda_{\max}(R_T)\le 0$6 occurs precisely when some branch, viewed as a Schrödinger operator $\lambda_{\max}(R_T)\le 0$7 on its line graph with absorbing boundary at $\lambda_{\max}(R_T)\le 0$8, has a potential well deep enough to bind a positive-energy state. This dichotomy is a feature not shared by adjacency matrices (whose spectral radius strictly increases under any pendant attachment) or Laplacians, where no analogous convergence-plus-sign-dependent-monotonicity phenomenon arises.

First-order expansion and tail monotonicity

Writing $\lambda_{\max}(R_T)\le 0$9 and $T$0, analytic perturbation theory applies when $T$1 is simple. With right/left eigenvectors $T$2 normalized by $T$3, the first-order coefficient is the spectral projection

$T$4

and

$T$5

The tail monotonicity theorem then states: if $T$6, the sequence $T$7 is eventually strictly monotonic—strictly decreasing toward $T$8 from above when $T$9, strictly increasing toward it from below when $\lambda_{\max}$0. The proof is elementary once the expansion is available: $\lambda_{\max}$1 retains the sign of $\lambda_{\max}$2 near $\lambda_{\max}$3, and $\lambda_{\max}$4 decreases monotonically to $\lambda_{\max}$5.

The simplicity assumption is handled explicitly: if $\lambda_{\max}$6 is multiple (e.g., two branches attain equal maxima, or coincide with the zero block), degenerate perturbation theory replaces $\lambda_{\max}$7 by $\lambda_{\max}$8, where $\lambda_{\max}$9 is the compression of $\lambda_{\max}(R_T)<0$0 to the eigenspace of $\lambda_{\max}(R_T)<0$1; the tail monotonicity conclusion is unchanged. The derivation for this case is deferred to Kato's classical treatment rather than carried out in full, which is the paper's principal expository gap.

Numerical verification

The worked example uses a base tree with $\lambda_{\max}(R_T)<0$2, one branch a path of length two and the other a binary fork. Both branch blocks are subcritical, $\lambda_{\max}(R_T)<0$3 and $\lambda_{\max}(R_T)<0$4, so $\lambda_{\max}(R_T)<0$5. Computing left and right eigenvectors for the zero eigenvalue gives $\lambda_{\max}(R_T)<0$6, predicting

$\lambda_{\max}(R_T)<0$7

with eventual strict decrease toward $\lambda_{\max}(R_T)<0$8 from above. The numerics confirm this and, notably, show that early behavior is genuinely non-monotonic: $\lambda_{\max}(R_T)<0$9, $R_T\in\mathbb{R}^{E\times E}$00, $R_T\in\mathbb{R}^{E\times E}$01, peaking near $R_T\in\mathbb{R}^{E\times E}$02 before decreasing through $R_T\in\mathbb{R}^{E\times E}$03. Thus the sequence crosses from negative to positive before settling into its predicted tail—a concrete demonstration that eventual monotonicity need not be global monotonicity. The complementary case $R_T\in\mathbb{R}^{E\times E}$04 is realized by the star $R_T\in\mathbb{R}^{E\times E}$05, where $R_T\in\mathbb{R}^{E\times E}$06 exactly, so $R_T\in\mathbb{R}^{E\times E}$07 and the sequence increases toward $R_T\in\mathbb{R}^{E\times E}$08 from below, matching the theorem.

Limitations and open questions

The theory rests on the affine structure $R_T\in\mathbb{R}^{E\times E}$09, which holds because only the degree of $R_T\in\mathbb{R}^{E\times E}$10 varies; it does not address simultaneous leaf addition at multiple vertices, where the perturbation parameter would couple several degrees. The degenerate-eigenvalue case is asserted by appeal to standard perturbation theory rather than proven in detail, and the paper does not characterize when $R_T\in\mathbb{R}^{E\times E}$11—in that case the first-order term vanishes and eventual monotonicity is undetermined by the present methods, requiring second-order analysis. Finally, the connection between the sign of $R_T\in\mathbb{R}^{E\times E}$12 and combinatorial features of the branch data (beyond the computed examples) remains unexplored.

Conclusion

This paper reduces the dynamics of $R_T\in\mathbb{R}^{E\times E}$13 under iterated leaf attachment to a finite-dimensional rank-one spectral perturbation problem. The limit $R_T\in\mathbb{R}^{E\times E}$14 depends only on Dirichlet-decoupled branch data and a universal zero floor contributed by the growing leaf cluster, while the sign of the projection coefficient $R_T\in\mathbb{R}^{E\times E}$15 governs both the approach direction and eventual strict monotonicity. Together with the earlier classification results, this yields a complete asymptotic picture of how local leaf addition shapes the spectrum governing discrete Einstein curvature on trees.