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

Toshiaki Omori¹ ; Hisashi Naito² ; Tatsuya Tate³

¹Tokyo University of Science
²Graduate School of Mathematics, Nagoya University
³Mathematical Institute, Tohoku University

The electronic journal of combinatorics, Tome 26 (2019) no. 3

Résumé

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.

Détail

DOI : 10.37236/8481

Classification : 05C50, 05C10, 52B05
Mots-clés : Goldberg-Coxeter construction