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

Toshiaki Omori, Hisashi Naito, Tatsuya Tate

研究成果: Article査読

抄録

We are concerned with spectral problems of the Goldberg-Coxeter construction for 3- and 4-valent finite graphs. The Goldberg-Coxeter constructions GCk,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(k2 + l2), nevertheless which have bounded girth. It is shown that the first (resp. the last) o(k2) eigenvalues of the combinatorial Laplacian on GCk,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 GCk,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 GC2k,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.

本文言語English
論文番号P3.7
ジャーナルElectronic Journal of Combinatorics
26
3
DOI
出版ステータスPublished - 2019

ASJC Scopus subject areas

  • 理論的コンピュータサイエンス
  • 幾何学とトポロジー
  • 離散数学と組合せ数学
  • 計算理論と計算数学
  • 応用数学

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