Eigenvalues of the Laplacian on the Goldberg-Coxeter Constructions for 3- and 4-valent Graphs

Abstract

We are concerned with spectral problems of the Goldberg-Coxeter construction for 3and 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.

A Short Lecture on Topological Crystallography and a Discrete Surface Theory

“…To discuss a convergence theory of trivalent discrete surfaces, we should consider how to subdivide a trivalent graph, and how to realize the subdivided graph. Goldberg-Coxeter subdivision for trivalent graphs defined by Dutour-Deza [7,8] is the good definition to subdivide a trivalent graph (see also Goldberg [11] and Omori-Naito-Tate [31]). Kotani-Naito-Omori [16], Tao [41], and Kotani-Naito-Tao [17] discuss convergences of sequences of trivalent discrete surfaces.…”

“…Eigenvalues of the Laplacian of graphs are also interest from physical and chemical view points (see also Section A.2). Some properties of eigenvalues of Laplacians of Goldberg-Coxeter constructions of trivalent graphs are discussed in Omori-Naito-Tate [31]. It is a very famous old result that the number of 2-dimensional space groups is 17.…”