Eigenvalues of the Laplacian on the Goldberg-Coxeter constructions for $3$- and $4$-valent graphs

TL;DR

This paper investigates the spectral properties of Goldberg-Coxeter constructions on 3- and 4-valent graphs, revealing how eigenvalues behave as the subdivision parameters grow, with implications for graph structure and eigenvalue multiplicities.

Contribution

It provides new asymptotic results for eigenvalues of Laplacians on Goldberg-Coxeter graphs and estimates relating these eigenvalues to the original graph's spectrum.

Findings

First eigenvalues tend to zero as subdivision increases.

Largest eigenvalues tend to 6 or 8 depending on valency.

Certain eigenvalues appear with high multiplicity regardless of initial graph structure.

Abstract

We are concerned with spectral problems of the Goldberg-Coxeter construction for $3$- and $4$-valent finite graphs. The Goldberg-Coxeter constructions $\mathrm{GC}_{k,l}(X)$ of a finite $3$- or $4$-valent graph $X$ are considered as "subdivisions" of $X$, whose number of vertices are increasing at order $O(k^2+l^2)$, nevertheless which have bounded girth. It is shown that the first (resp. the last) $o(k^2)$ eigenvalues of the combinatorial Laplacian on $\mathrm{GC}_{k,0}(X)$ tend to $0$ (resp. tend to $6$ or $8$ in the $3$- or $4$-valent case, respectively) as $k$ goes to infinity. A concrete estimate for the first several eigenvalues of $\mathrm{GC}_{k,l}(X)$ by those of $X$ is also obtained for general $k$ and $l$. It is also shown that the specific values always appear as eigenvalues of $\mathrm{GC}_{2k,0}(X)$ with large multiplicities almost independently to the structure of the initial $X$. In contrast, some dependency of the graph structure of $X$ on the multiplicity of the specific values is also studied.