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Effect of edge-stretching on Steklov eigenvalues and sharp Steklov eigenvalue bounds on leaf--boundary trees

Jiangdong Ai, Yizhe Ji, Xiaopan Lian, Kun Yang

Abstract

Let $T$ be a finite tree with leaf set $\dO$ as the boundary and let $λ_2$ be the first nontrivial Steklov eigenvalue. Let $D$ and $\ell$ be the maximum vertex degree and the number of leaves, respectively. Motivated by the spectral influence of neck-stretching on Riemannian manifolds, we investigate a discrete counterpart--edge-stretching--and its effect on the Steklov eigenvalues of graphs. We prove that Steklov eigenvalues decrease monotonically under the edge--stretching operation. As a consequence, we prove that $λ_2\le D/\ell$, with equality if and only if $T$ is a star. This fundamentally improves the constant in He--Hua's bound $λ_2\le 4(D-1)/\ell$ to the optimal value~$1$. We also provide a closed-form diagonalization of the Steklov problem on level--regular trees, yielding explicit eigenvalues and multiplicities. In addition, we provide a general upper bound $λ_k\le \min\{1,\,16Dk/\ell\}$ for higher eigenvalues. Systematic numerical experiments verify the sharp bound and provide evidence for the extremal conjecture of Lin--Zhao on balanced minimum--height trees.

Effect of edge-stretching on Steklov eigenvalues and sharp Steklov eigenvalue bounds on leaf--boundary trees

Abstract

Let be a finite tree with leaf set as the boundary and let be the first nontrivial Steklov eigenvalue. Let and be the maximum vertex degree and the number of leaves, respectively. Motivated by the spectral influence of neck-stretching on Riemannian manifolds, we investigate a discrete counterpart--edge-stretching--and its effect on the Steklov eigenvalues of graphs. We prove that Steklov eigenvalues decrease monotonically under the edge--stretching operation. As a consequence, we prove that , with equality if and only if is a star. This fundamentally improves the constant in He--Hua's bound to the optimal value~. We also provide a closed-form diagonalization of the Steklov problem on level--regular trees, yielding explicit eigenvalues and multiplicities. In addition, we provide a general upper bound for higher eigenvalues. Systematic numerical experiments verify the sharp bound and provide evidence for the extremal conjecture of Lin--Zhao on balanced minimum--height trees.

Paper Structure

This paper contains 13 sections, 15 theorems, 35 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Proofs
  3. Preliminaries
  4. Edge stretching
  5. Explicit spectra for level--regular trees
  6. Sharp upper bound for lambda_2
  7. General lambda_k bound
  8. Numerical experiments
  9. Exhaustive verification of the sharp bound
  10. Exhaustive search in TS(l,D)
  11. Randomized experiments for larger leaf counts
  12. Reproducibility.
  13. Concluding remarks

Key Result

Theorem 1.2

Let $G^{(L)}$ be obtained from $(G,\delta\Omega)$ by $L$--stretching a single edge. Then for every $k=1,\dots,|\delta\Omega|$,

Figures (4)

  • Figure 1: Neck stretching obtained by inserting a cylindrical region $[0,L]\times\Sigma$.
  • Figure 2: The $3$--stretch of an edge $e=(u,v)$: the edge is replaced with a path of length $3$ through new interior vertices $w_1,w_2$ (gray).
  • Figure 3: Level--regular trees with non-constant branching sequences. Left: $\mathbf{m}=(3,2)$, $6$ leaves. Right: $\mathbf{m}=(2,3)$, $6$ leaves. Both have $6$ leaves but different spectra (see Example \ref{['ex:nonconstant']}).
  • Figure 4: The two key trees in $\mathrm{TS}(8,3)$. Interior vertices are filled (black); leaves (boundary) are open (white). The unbalanced tree $T_1$ (left) achieves a larger $\lambda_2$ than the balanced tree $T_b^*$ (right).

Theorems & Definitions (36)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Definition 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Lemma 2.1: Harmonic extension minimizes energy
  • proof
  • ...and 26 more