How the eigenvalue region of standardized Laplacians leaves the 2005 polygon
Abstract
Let Λn be the set of all eigenvalues of n× n standardized Laplacian matrices, and let Sn⊆Λn be the polygon introduced in 2005, with upper vertices λ0=0,…,λn−1=1. It is now known that Λn=Sn, and that Λn is convex, if and only if n≤4. We study how Λn leaves Sn near an edge [λm,λm+1], 2(m+1)<n. First, we prove a switching law for matrices having a boundary point of Λn as a simple eigenvalue: an explicit switching function on the arcs, built from the left and right eigenvectors and the outward normal, forces maximal weight where positive and zero weight where negative; intermediate weights occur only where it vanishes. Second, we perturb the weights of the arcs k→ k+m+1 in the circulant digraph (arcs k→ k+s, 1≤ s≤ m, plus these arcs with weight τ) that realizes the point λm+τ(λm+1−λm). At second order, the first Fourier harmonic of the perturbation moves the eigenvalue beyond the line of the edge if and only if τ<τ*m=sin(πm/n)/(2cos(π/n)sin(π(m+1)/n)). For even n and m, the point with τ=τ*m is the vertex of Sn/2 lying on this edge, and digraphs with these arcs only at even k give an explicit eigenvalue path that leaves Sn at λm and, in all computed cases, ends at this vertex. Third, we give a conjectural, numerically supported picture of ∂Λ5: beyond each edge of S5 not incident to $0$ or $1$, Λ5 forms a cap bounded by four algebraic arcs, which leaves the edge tangentially at its endpoint adjacent to $0$ or $1$ and rejoins it at τ*1=(3−√5)/2; the rest of the edge lies on ∂Λ5. In computations for n≤8, a cap returns to its edge smoothly at τ*m, at a corner at τ*m, or at a corner beyond τ*m.
Related
- Metric dimension and maximum eigenvalue multiplicity of graph Laplacians
- The Eigenvalue Region of a Family of Markov Chains with Randomness Confined to Two States
- Numerical results for the Stokes problem and the iterated Laplacian in a polygon
- Generalized Dirichlet Energy and Graph Laplacians for Clustering Directed and Undirected Graphs
- Strict rearrangement inequalities for logarithmic p-Laplacians
BibTeX
@misc{chebotarev2026eigenvalue,
title = {{How the eigenvalue region of standardized Laplacians leaves the 2005 polygon}},
author = {Pavel Chebotarev},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02249},
eprint = {2611.02249},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02249},
}
RIS
TY - UNPB
TI - How the eigenvalue region of standardized Laplacians leaves the 2005 polygon
AU - Pavel Chebotarev
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.02249
UR - https://arxiv.org/abs/2611.02249
AB - Let Λ_n be the set of all eigenvalues of n× n standardized Laplacian matrices, and let S_n⊆Λ_n be the polygon introduced in 2005, with upper vertices λ_0=0,…,λ_n−1=1. It is now known that Λ_n=S_n, and that Λ_n is convex, if and only if n≤4. We study how Λ_n leaves S_n near an edge [λ_m,λ_m+1], 2(m+1)<n. First, we prove a switching law for matrices having a boundary point of Λ_n as a simple eigenvalue: an explicit switching function on the arcs, built from the left and right eigenvectors and the outward normal, forces maximal weight where positive and zero weight where negative; intermediate weights occur only where it vanishes. Second, we perturb the weights of the arcs k→ k+m+1 in the circulant digraph (arcs k→ k+s, 1≤ s≤ m, plus these arcs with weight τ) that realizes the point λ_m+τ(λ_m+1−λ_m). At second order, the first Fourier harmonic of the perturbation moves the eigenvalue beyond the line of the edge if and only if τ<τ^*_m=sin(πm/n)/(2cos(π/n)sin(π(m+1)/n)). For even n and m, the point with τ=τ^*_m is the vertex of S_n/2 lying on this edge, and digraphs with these arcs only at even k give an explicit eigenvalue path that leaves S_n at λ_m and, in all computed cases, ends at this vertex. Third, we give a conjectural, numerically supported picture of ∂Λ_5: beyond each edge of S_5 not incident to $0$ or $1$, Λ_5 forms a cap bounded by four algebraic arcs, which leaves the edge tangentially at its endpoint adjacent to $0$ or $1$ and rejoins it at τ^*_1=(3−√5)/2; the rest of the edge lies on ∂Λ_5. In computations for n≤8, a cap returns to its edge smoothly at τ^*_m, at a corner at τ^*_m, or at a corner beyond τ^*_m.
ER -