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.
ASJC Scopus subject areas
- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics
- Applied Mathematics