TY - JOUR

T1 - Eigenvalues of the Laplacian on the Goldberg-Coxeter constructions for 3- and 4-valent graphs

AU - Omori, Toshiaki

AU - Naito, Hisashi

AU - Tate, Tatsuya

N1 - Funding Information:
Authors are partially supported by JSPS KAKENHI Grant Number 15K17546, 15H02055, 25400068, 18K03267, 26400067, 17H06465, 17H06466, and 19K03488. This work is also supported by JST CREST Grant Number JPMJCR17J4.
Publisher Copyright:
© The authors.

PY - 2019

Y1 - 2019

N2 - 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.

AB - 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.

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U2 - 10.37236/8481

DO - 10.37236/8481

M3 - Article

AN - SCOPUS:85071273744

VL - 26

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

SN - 1077-8926

IS - 3

M1 - P3.7

ER -