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Eigenvalues of the Laplacian on the Goldberg-Coxeter constructions for $3$- and $4$-valent graphs
- Source :
- The Electronic Journal of Combinatorics, 26(3) (2019), #P3.7
- Publication Year :
- 2018
-
Abstract
- We are concerned with spectral problems of the Goldberg-Coxeter construction for $3$- and $4$-valent finite graphs. The Goldberg-Coxeter constructions $\mathrm{GC}_{k,l}(X)$ of a finite $3$- or $4$-valent graph $X$ are 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 $\mathrm{GC}_{k,0}(X)$ 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 for the first several eigenvalues of $\mathrm{GC}_{k,l}(X)$ 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 $\mathrm{GC}_{2k,0}(X)$ 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.<br />Comment: 23 pages, 11 figures, authors' final version (to appear in The Electronic Journal of Combinatorics)
Details
- Database :
- arXiv
- Journal :
- The Electronic Journal of Combinatorics, 26(3) (2019), #P3.7
- Publication Type :
- Report
- Accession number :
- edsarx.1807.10891
- Document Type :
- Working Paper