### Abstract

We are concerned with spectral problems of the Goldberg-Coxeter construction for 3- and 4-valent finite graphs. The Goldberg-Coxeter constructions 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 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 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 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.

Original language | English |
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Article number | P3.7 |

Journal | Electronic Journal of Combinatorics |

Volume | 26 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2019 Jan 1 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics
- Applied Mathematics

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## Cite this

*Electronic Journal of Combinatorics*,

*26*(3), [P3.7]. https://doi.org/10.37236/8481