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We are concerned with spectral problems of the Goldberg-Coxeter constructionfor $3$- and $4$-valent finite graphs. The Goldberg-Coxeter constructions$\mathrm{GC}_{k,l}(X)$ of a finite $3$- or $4$-valent graph $X$ are consideredas "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 forthe first several eigenvalues of $\mathrm{GC}_{k,l}(X)$ by those of $X$ is alsoobtained for general $k$ and $l$. It is also shown that the specific valuesalways appear as eigenvalues of $\mathrm{GC}_{2k,0}(X)$ with largemultiplicities almost independently to the structure of the initial $X$. Incontrast, some dependency of the graph structure of $X$ on the multiplicity ofthe specific values is also studied.

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