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

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.


Introduction
The Goldberg-Coxeter construction is a subdivision of a 3-or 4-valent graph, and it is defined by Dutour-Deza [4] for a plane graph based on a simplicial subdivision of regular polytopes in [1,7]. In [4], it is pointed out that it often appears in chemistry and architecture, and its combinatorial and algebraic structures are investigated. Goldberg-Coxeter constructions of regular polyhedra generate a class of Archimedean polyhedra, and infinite sequence of polyhedra, which are called Goldberg polyhedra. For example a Goldberg-Coxeter construction of a dodecahedron generates a truncatedicosahedron, which is known as a fullerene C 60 [10,17]. Goldberg-Coxeter constructions are also applied to Mackay-like crystals, and explain large scale of spatial fullerenes [14,16]. Mathematical modeling of self-assembly in nature is also widely studied in [1,11]. Recently, Fujita et. al. have synthesized molecule structures with 4-valent Goldberg polyhedra, and they explain self-assembly from viewpoints of chemistry and biology [6].
On the other hand, the stability of a molecule is explained by eigenvalues of the finite graphs which express the molecule structure by Hückel method [2]. Hence, studies for eigenvalues of Goldberg-Coxeter constructions are worth trying. The Goldberg-Coxeter construction GC k,l (X) of a 3-or 4-valent graph X has the parameters k and l both of which are integers and they are regarded to indicate a point in the triangular or square lattices, respectively. Then we are concerned with behavior of eigenvalues of GC k,l (X) when k and l tends to infinity.
Throughout this paper, unless otherwise indicated, a graph is always assumed to be connected, finite and simple. For a graph X, let us denote by V(X) the set of vertices of X, and by E(X) the set of undirected edges of X. For p ∈ V(X), the set of its neighboring vertices is denoted by N X (p). The combinatorial Laplacian ∆ X , simply called the Laplacian, of a graph X acts on the set R V(X) of functions on V(X) and is defined as (X) and p ∈ V(X), where deg(p) = 3 or 4 provided X is respectively a 3-or 4-valent graph. As is well-known, the eigenvalues of ∆ X for a regular graph X of degree r necessarily lie in the interval [0, 2r].
The definition of the Goldberg-Coxeter constructions extends for general 3-or 4-valent graph X = (V(X), E(X)) equipped with an orientation at each vertex, in the sense that, for each p ∈ V(X), the set of edges emanating from p is ordered, and the following is proved. Theorem 1.1. Let X = (V(X), E(X)) be a connected, finite and simple 3or 4-valent graph equipped with an orientation at each vertex, X = GC k,l (X) be the Goldberg-Coxeter construction of X, where k ≥ l ≥ 0 and k 0 and 0 = λ 1 (X) < λ 2 (X) ≤ · · · ≤ λ |V(X)| (X), 0 = λ 1 (X ) < λ 2 (X ) ≤ · · · ≤ λ |V(X )| (X ) be the eigenvalues of their Laplacians ∆ X , ∆ X , respectively. Then there exist integers µ(k, l) and ν(k, l) depending only on k and l satisfying for i = 1, 2, . . . , |V(X)|. When X is 3-valent, µ(k, l) satisfies When X is 4-valent, ν(k, l) satisfies As shall be explained later (cf. Proposition 2.2), if, in particular, X is "appropriately" embedded in an oriented surface, then X is endowed with a natural orientation at each vertex and GC k,l (X) remains to be also embedded in the same surface. Thus (1.1) also gives an upper bound for such a graph X.
There is a long line of works on upper bounds for the (especially, first nonzero) eigenvalues of general planar or genus g finite graphs (see [12,18] and the references therein). In [13], it is proved that the i-th eigenvalue of a graph embedded in an oriented surface of genus g is estimated from above by O((g + 1) log 2 (g + 1)i/n), where n is the number of the vertices. Our estimate (1.1) is different from their estimate on the point that (1.1) is independent of the genus.
On the other hand, as for the last several eigenvalues of GC k,0 (X) the following holds.
for i = 1, 2, . . . , |V(X)|. If X is a bipartite 3-valent graph, then the convergence (1.3) remains valid also for arbitrary GC k,l (X). Furthermore, for a fixed k, the last |V(X)| eigenvalues of n-th iterated Goldberg-Coxeter constructions GC n k,0 (X) converge to 6 or 8 exponentially fast as n → ∞. As the following theorems show the Goldberg-Coxeter constructions have also steady eigenvalues. Theorem 1.3. Let X be a connected, finite and simple 3-valent graph equipped with an orientation at each vertex, and GC 2k,0 (X) be its Goldberg-Coxeter constructions for k ∈ N.
Here x is the smallest integer ≥ x, and x is the largest integer ≤ x.
Theorem 1.4. Let X be a connected, finite and simple 4-valent graph equipped with an orientation at each vertex, and GC 2k,0 (X) be its Goldberg-Coxeter constructions for k ∈ N.
Problems on eigenvalues of combinatorial Laplacian on regular graphs are extensively investigated. In particular, an explicit formula of a limit density of eigenvalue distributions of certain sequences of regular graphs was obtained in [15], and its geometric proof using a trace formula is given in [9] (see also [3]). One of points in these works is that the sequence {X n } of q-regular graphs with number of vertices |X n | → ∞ as n → ∞ is assumed to have large girths g(X n ) → ∞ as n → ∞. From this assumption, the graphs X n get similar, as n → ∞, to a universal covering graph, namely a q-regular tree at least locally, and then a trace formula becomes able to apply. The girths of the Goldberg-Coxeter constructions {GC k,l (X)} k,l with an initial graph X are uniformly bounded with respect to the parameters k and l, and hence it would not be so straightforward to apply a trace formula to obtain a limit distribution of the eigenvalue distributions. Indeed, from several numerical results it is considered that the limit distributions of eigenvalue distributions of Goldberg-Coxeter constructions is not quite universal. This speculation is also supported by the following results.
Theorem 1.5. Let X be a connected, finite and simple 3-valent graph which is embedded in a plane. Assume that the number of edges surrounding each face is divisible by 3. Then the following hold.
(2) For any k ∈ N, both GC k,0 (X) and GC k,k (X) have eigenvalue 4 (resp. 2), whose multiplicity is at least k/2 (resp. k/2 ). This paper is organized as follows. In Section 2, after giving the precise definition of the Goldberg-Coxeter constructions GC k,l (X), let us study their structure which is related with the spectral problems. In Section 3, we obtain some comparisons of the eigenvalues of X and GC k,l (X) to prove Theorem 1.1. In Section 4, we first present proofs of Theorem 1.3 and 1.4. At the end of this paper, we shall give a few criteria for a 3-valent plane graph X so that some GC k,0 (X)'s have eigenvalues 2 or 4, which proves Theorem 1.5.

Goldberg-Coxeter constructions
This section studies the structure of Goldberg-Coxeter constructions, which shall be necessary in the subsequent sections.
The notion of Goldberg-Coxeter constructions is defined, due to Deza-Dutour [4,5], for a plane graph. The definition can extend for a nonplanar graph X; indeed, X has only to be equipped with an "orientation at each vertex", and if, in particular, X is "appropriately" embedded on an oriented surface, then the constructions can be done on the surface (see Proposition 2.4 gives the triangular lattice on C having 0, 1 and ω as its fundamental triangle, while Z[i] gives the square lattice on C having 0, 1, 1 + i and i as its fundamental square. Definition 2.1 (cf. Deza-Dutour [4,5]). Let X be a connected, finite and simple 3-or 4-valent (abstract) graph equipped with an orientation at each vertex in the sense that, for each p ∈ V(X), the set of edges emanating from p is ordered. For (k, l) ∈ Z 2 , (k, l) (0, 0), the Goldberg-Coxeter construction of X with parameters k and l is defined through the following steps.
(i) Let us first consider the equilateral triangle = (0, z, ωz) in Z[ω] having the vertices 0, z = k + lω and ωz (resp. the square = (0, z, (1 + i)z, iz) in Z[i] having the vertices 0, z = k + li, (1 + i)z and iz). (ii) Let us take all the small triangles in Z[ω] (resp. squares in Z[i]) intersecting with (resp. ) in its interior and join the barycenters of the neighboring small triangles (resp. squares) to obtain a graph, which is, as an associated (abstract) graph with p for each p ∈ V(X), denoted by (p) (resp. (p)). Let us assign each of the edges emanating from p to exactly one edge of the triangle (resp. square) so that the orientation at p coincides with the standard orientation of in Z[ω] (resp. in Z[i]). Note that (p) (resp. (p)) has the 2π/3-rotational symmetry (resp. the π/2-rotational symmetry).
Proposition 2.2. Let X be a connected, finite and simple 3or 4-valent graph which is embedded in an oriented surface M in such a way that the closure of each face is simply connected. Then for (k, l) ∈ Z 2 , (k, l) (0, 0), GC k,l (X) is well-defined and is also embedded in M.
Proof. The oriented tangent plane to M at p ∈ V(X) defines the orientation at p, and GC k,l (X) is defined. The notion of faces is also well-defined. Since each face of X is simply connected, we can take a dual graph D X of X in M, all of whose faces are simply connected triangles (resp. rectangles) for the 3-valent case (resp. 4-valent case). The dividing step (ii) and the gluing step (iii) in Definition 2.1 are well done in M via respective appropriate local charts.
A Goldberg-Coxeter construction GC k,l (X) for 3-valent (resp. 4-valent) graph X inserts some hexagons (resp. squares), according to its parameter k and l, between each pair of original faces of X. The most famous example is a fullerene C 60 , called also a buckminsterfullerene or a buckyball, which is nothing but GC 1,1 (Dodecahedron). This construction owes its name to the pioneering work [7] due to M. Goldberg, where a so-called Goldberg polyhedron (a convex polyhedron whose 1-skeleton is a 3-valent graph, consisting of hexagons and pentagons with rotational icosahedral symmetry 3-valent graph as its 1-skeleton) is studied and is proved to be of the form GC k,l (Dodecahedron) for some k and l. A Goldberg-Coxeter construction for 3-or 4-valent plane graphs occurs in many other context; see [4] and the references therein. Several examples of Goldberg-Coxeter constructions for nonplanar 3-valent (infinite or finite quotient) graphs, such as for carbon nanotubes and Mackay-like crystals, are provided in [14].
The following proposition summarizes a few fundamental properties of Goldberg-Coxeter constructions. [4,5]). Let X = (V(X), E(X)) be a 3-valent (resp. 4-valent) graph equipped with an orientation at each vertex. Then the following hold.

Proposition 2.3 (Deza-Dutour
(1) If X is embedded in an oriented surface in such a way that the closure of each face is simply connected, and the orientation at each vertex coincides with the one of the surface, then GC z (GC z (X)) = GC zz (X), for any z, z ∈ Z[ω] (resp. z, z ∈ Z[i]). (2) For any (k, l) ∈ Z 2 , (k, l) (0, 0), we have the following graph isomorphisms: In consideration of Proposition 2.3 (2), in the rest of this paper, we assume that k is a positive integer and l is a nonnegative integer satisfying k ≥ l ≥ 0 and k 0.

Clusters for Goldberg-Coxeter constructions.
A cluster is the central notion in this paper. Its definitions shall be given below in two different cases: where X is 3-valent and where X is 4-valent.
, called the (k, l)-cluster, so as to have k 2 + kl + l 2 vertices and the 2π/3-rotational symmetry of (p). For this, we just have to define V(p) by the set of vertices x of (p) (considered as the graph on ⊆ Z[ω]) satisfying one of the following conditions: (i) x ∈ (p) corresponds to a triangle in Z[ω] whose barycenter lies in the interior of = (0, z, ωz), where z = k + lω; (ii) x ∈ (p) corresponds to an upward triangle in Z[ω] whose barycenter lies on an edge of . Here we mean an upward triangle (a, b) by the triangle in Z[ω] with vertices a + bω, a + 1 + bω and a + (b + 1)ω for a, b ∈ Z (see Figure 1). We also denote by (a, b), called downward triangle, the triangle with vertices a + bω, a + (b + 1)ω and a − 1 In the case that l = 0, X(p) is nothing but (p) itself, has k 2 vertices and has the dihedral symmetry D 3 (of order 6) (see Figure 1).
In the case that k = l > 0, it is easily seen that there are 3(k 2 − k) vertices satisfying (i) and 3k vertices satisfying (ii). The obtained subgraph X(p) has 3k 2 vertices and has the 2π/3-rotational symmetry because upward triangles are mapped to upward triangles by the rotation (see Figure 1).
The following lemma makes clear the cases where there is a barycenter lying on an edge of among the remaining cases.
Moreover, in the case above, each edge of passes through exactly 2m = 2 gcd(k, l) barycenters. Among these 2m vertices, exactly m vertices corresponding to upward triangles have just two adjacent triangles with barycenters lying in . The combined 3m vertices on the three edges of are located in symmetric position with the rotation by 2π/3 of .
Lemma 2.4 shows that the subgraph X(p) has (k − l) 2 + 3kl = k 2 + kl + l 2 vertices and also has the 2π/3-rotational symmetry in the remaining case that k > l > 0.
Here we can prove the following proposition, which guarantees that the bipartiteness is kept after a Goldberg-Coxeter construction.
Proposition 2.5. Let X be a 3-valent bipartite graph equipped with an orientation at each vertex. Then for any (k, l) ∈ Z 2 , (k, l) (0, 0), GC k,l (X) is also bipartite. So the spectrum of GC k,l (X) is symmetric with respect to 3.
Proof. Let a bipartition of X be given and either black or white be assigned to each vertex p ∈ V(X).
Each vertex x of each (k, l)-cluster X(p) can be colored according to a rule that if p is white, then • paint x black, provided the triangle in Z[ω] corresponding to x is upward; • paint x white, provided the triangle in Z[ω] corresponding to x is downward; and if p is black, then exchange black and white above. A white vertex is adjacent only to black vertices in X, and two adjacent clusters X(p) and X(q) are positioned, in Z[ω], at π-rotation around the midpoint of an edge of , which switches upward and downward triangles. So, the rule above gives a bipartition of GC k,l (X).
Similarly as in the 3-valent case, we construct for each p ∈ V(X) an appropriate subgraph X(p) = (V(p), E(p)) of (p), still called the (k, l)-cluster, so as to have k 2 + l 2 vertices. To this end, we need to clarify the cases where a barycenter of a small square in Moreover, if this is the case, each edge of passes through exactly m barycenters.
Unlike the 3-valent case, we cannot choose a cluster X(p) with k 2 + l 2 vertices to have the π/2rotational symmetry in the case where k 1 0 (mod 2), k 1 ≡ l 1 (mod 2) and m 0 (mod 2) because no vertex of (p) is positioned at the barycenter of and k 2 + l 2 = m 2 ((k 1 − l 1 ) 2 + 2k 1 l 1 ) is not divided by 4. Even in such cases, X(p) only has to have the same number of outward edges among the four directions to every adjacent cluster.
Lemma 2.7. Let X be a 4-valent graph equipped with an orientation at each vertex. Then there exists an Euler circuit ε of X which turns either left or right at every vertex of X.
Proof. As is well-known, any 4-valent graph X has an Euler circuit, which is by definition a closed path in X which visits every edge exactly once. Let us take an Euler circuit ε of X and suppose that ε goes straight ahead at a vertex p ∈ V(X). The circuit ε comes back to p again from one of the other directions after it straight ahead at p (because X is 4-valent). By following the interval in opposite directions, the obtained circuit goes straight ahead one time fewer than ε. This proves Lemma 2.7.
The Euler circuit ε obtained in Lemma 2.7 assigns a direction to each edge of X such that the direction alternates between inward and outward at each vertex of X.
Now we can clearly define V(p) by the set of vertices x of (p) satisfying one of the following conditions: (i) x corresponds to a square in Z[i] whose barycenter lies in the interior of ; (ii) x corresponds to a barycenter lying on the two edges of with opposite sides which correspond to the outward edges of X with respect to the Euler circuit ε in Lemma 2.7. (C) V(X) can be colored by two colors, say black and white, with the following properties: (C-i) A black vertex is adjacent to three white vertices; (C-ii) a white vertex is adjacent to exactly one black vertex, so the other two adjacent vertices are white; (C-iii) for any pair of black vertices x, y ∈ V(X) which are three vertices away from each other, there is a path from x to y either turning left twice or turning right twice.
The coherent edge numbering (CN) implies the condition (N); indeed, let p ∈ V(X) and let e 1 , e 2 and e 3 be three edges of X emanating from p. We assign 0 to p regarded as a vertex of GC 2,0 (X), and, for i = 1, 2 and 3, assign i to the vertex of GC 2,0 (X) positioned at the "opposite-side" to e i . The resulting numbering of vertices of GC 2,0 (X) satisfies (N-i) and (N-ii) (see Figure 3). Moreover, as is easily proved, (N) implies the condition (F). So the following proposition shows that (F), (CN) and (N) are mutually equivalent.  where H 1 (X, Z) is the 1-dimensional homology group of X. Now any γ ∈ H 1 (X, Z) can be written as γ = f : face of X a f ∂ f , where a f ∈ Z and ∂ f is the cycle consisting of edges around f . Our assumption implies that ϕ(∂ f ) = 3 for any face f of X. Hence we conclude that ϕ ≡ 3, which implies that ϕ ≡ 3 on CP(X, e 0 ).

A relation between (F) and (C) is stated as follows.
Proposition 2.9. Let X be a 3-valent plane graph satisfying (F). Then X has a vertex coherent coloring satisfying (C-i), (C-ii) and (C-iii).
Proof. let p 0 ∈ V(X) be an arbitrary fixed vertex and color it black. Every vertex which is accessible by either turning left twice or turning right twice from a black vertex is, one after another, colored in black until no more vertices can be colored in black. The remaining vertices are colored in white. Now we have to check that (C-i) and (C-ii) are satisfied (while (C-iii) is necessarily satisfied). It is easily seen that a white vertex is adjacent to at least one black vertex; otherwise, all vertices of X must be white. It is also easily checked that if a white vertex is adjacent to two or more black vertices, then two other black vertices are necessarily adjacent somewhere else. So, it suffices to show that any pair of black vertices cannot be adjacent. Suppose that there is a pair of adjacent black vertices, say p, q ∈ V(X). From our way of the coloring, there is a path γ from p to q which is a sequence of either twice turning left or twice turning right between black vertices. Then γ ∪ (q, p) is a closed path, which surrounds a finitely many faces, say f 1 , f 2 , . . . , f n , after removing back-trackings. Now if n = 1, then γ consists of a circuit on the boundary ∂ f 1 of a face f 1 and of some back-trackings with black base points on ∂ f 1 , which is a contradiction because the total of τ defined by (2.3) is 0 (mod 3) after the crossing just prior to a lap of γ ∪ (q, p). So assume that n ≥ 2. There are just two possibilities of paths along the boundary of n i=1 f i connecting a pair of black vertices with distance 3, as indicated in Figure 4. In either case, we can replace γ ∪ (q, p) by a closed path which does not surround a face f i (by ignoring back-trackings), and is still a sequence of either twice turning left or twice turning right between black vertices. Therefore the conclusion for the case where n ≥ 2 can be deduced from the discussion given for the case n = 1.

Examples 2.10.
(1) The tetrahedron and any of its Goldberg-Coxeter constructions satisfy all the conditions above. (2) GC 2,0 (X) for any 3-valent plane graph X always satisfies (C-i), (C-ii) and (C-iii); indeed, we just have to color only the "center" of each (2, 0)-cluster black, and the others white. (3) GC 1,1 (X) for any 3-valent plane graph X also always satisfies (C-i), (C-ii) and (C-iii); indeed, we just have to color in accordance with the rule shown in Figure 6. 3.1. The case where X is 3-valent. Let p ∈ V(X), q ∈ N X (p) and set . Note that, for any x ∈ V(p) and q ∈ N X (p), there are at most two edges emanating from x to V(q).
Proof of Theorem 1.1. Since there is nothing to discuss when (k, l) = (1, 0), we only consider the other cases. Let c = 1/ |V(p)| = 1/ √ k 2 + kl + l 2 and define a linear map Q : (X) and for x ∈ V(p) by The transpose t Q : R V(X ) → R V(X) of Q is then written as for g ∈ R V(X ) and p ∈ V(X). It then follows that for any f ∈ R V(X) and for any p ∈ V(X), The second term equals −3c 2 |V 0 (p)| f (p) and the third term is computed as where the last equality follows from the symmetry of X(p). Therefore we obtain where µ(k, l) is the number of edges in X connecting two clusters and depends only on k and l.
(1.1) of Theorem 1.1 now immediately follows from the following.
Theorem 3.1 (Interlacing property, see for example [2]). Let Q be a real n × m matrix satisfying t QQ = I m and A be a real symmetric n × n matrix. If the eigenvalues of A and t QAQ are The equality (1.2) for l = 0 or k = l > 0 are easily proved. Let us estimate the number of edges crossing the edge E = 0z when k > l > 0. Notice first that there is at most one crossing edge emanating from an upward triangle (a, b), and that there are at most two crossing edge emanating from a downward triangle (a, b). For each c ∈ Z, "the zigzag path" which is obtained by joining the barycenters of (a, b), (a, b) and (a + 1, b − 1) for all a, b ∈ Z with a + b = c crosses the edge E = 0z exactly once provided 0 ≤ c ≤ k + l − 1 and does not cross E otherwise. Also, the line passing through a ∈ Z with slant 1 + ω crosses E exactly once provided 0 ≤ a ≤ k − l and does not cross E otherwise. Therefore the number of edges crossing E is at most k + l + (k − l − 2) = 2k − 2. (See Figure 7 for an example. ) Figure 7. (k, l) = (9, 3). 15 edges cross the edge E = 0z, (z = 9 + 3ω).
The assertion in Theorem 1.2 for a bipartite graph X is an immediate consequence from Theorem 1.1 and Proposition 2.5. The former one in Theorem 1.2 follows from the following.  The following remark shall be repeatedly used in the sequel: by assigning the same function u to the other clusters, we have a global function u : GC k,0 (X) → R, which is an eigenfunction of ∆ GC k,0 (X) with eigenvalue λ; indeed, (i) ∆ X(p) u = λu is equivalent to a Neumann problem: Theorem 3.2 is an immediate consequence from the the following Lemmata 3.4 and 3.5.
Proof. Let us replace c in (3.2) by u : GC k,0 (X) → R which is obtained from a D 3 -invariant eigenfunction on the (k, 0)-cluster. We may assume that x∈V(p) u(x) 2 = 1/|V(X)|, so that t QQ = id R V (X) . After a straightforward computation using (i) and (ii) in Definition 3.3 for u, we can obtain the following equality: for any f ∈ R V(X) and any p ∈ V(X), where q ∈ N X (p) is an adjacent vertex to p. Again from Theorem 3.1, the desired inequality is proved. Proof. Let us first construct all the eigenfunctions on a hexagonal lattice with toroidal boundary condition. If we set m := (1 + ω)/3, where ω = e πi/3 , then the discrete set is naturally regarded as a hexagonal lattice. For a fixed k ∈ N, let us consider the equations for a function v on the parallelogram where a and b in (3.7) are considered modulo k, such as for the former equation of (3.7) with a = b = 0. So if v solves (3.7), then it gives an eigenfunction with eigenvalue λ on the finite 3-valent graph T (k) with 2k 2 vertices obtained by adding edges between a and m + a + (k − 1)ω, and between bω and m + k − 1 + bω for each a, b = 0, 1, . . . , k − 1.
We now claim that (the real part of) the average u := σ∈D 6 σv + 1,0 under an action of D 6 on T (k) gives a function on the (k, 0)-cluster {a+bω ∈ P(k) | a+b ≤ k−1}∪{m+a+bω ∈ P(k) | a+b ≤ k−2} satisfying (i) and (ii) in Definition 3.3 with λ = λ + 1,0 . Here D 6 is a dihedral group of order 12 generated by the three automorphisms on T (k) induced from • the rotation by 2π/3: • the reflection along a diagonal line of the parallelogram: • and the reflection along the other one: where a and b are again considered modulo k and these maps are considered as P(k) → P(k). Since the Laplacian on a graph is equivariant under the action of an automorphism and the Neumann boundary condition as in (3.3) is satisfied by the definition of u, u satisfies ∆ X(p) u = λ + 1,0 u. Moreover it is easily checked by computing the total sum of v + 1,0 along the "boundary" of P(k) and the "diagonal line between (m+) k − 1 and (m+) (k − 1)ω" of P(k) that u is not identically zero (except in the case k = 1). This proves that λ + 1,0 (k) is a D 3 -invariant eigenvalue for the (k, 0)-cluster (k ≥ 2). Since λ ± s,t = 6 if and only if (s, t) = (0, 0) and the sign is positive, and since a D 3 -invariant eigenvalue is necessarily an eigenvalue of T (k), λ + 1,0 above is the largest D 3 -invariant eigenvalue for the (k, 0)-cluster.
3.2. The case where X is 4-valent. The same notation as (3.1) is used also in the 4-valent case. The notion of D 4 -invariant eigenvalue is also defined exactly in the same way as in the 3-valent case. Since the proof of Theorem 1.1 for the 4-valent case is almost similar as that for the 3-valent case, we omit it. Moreover, (3.4) is valid also for a 4-valent graph; indeed, the same equality as in (3.5) holds, whose proof is also omitted. A corresponding result to Lemma 3.5 is stated as follows (its proof is omitted again).
Lemma 3.7. Let k be an integer with k ≥ 2. If k is even (resp. odd), then λ = 4 + 4 cos 2π k resp. 4 + 4 cos π k is the second largest (resp. the largest) D 4 -invariant eigenvalue for the (k, 0)-cluster and converges to 8 as k tends to infinity.

On the eigenvalues 2 and 4 for Goldberg-Coxeter constructions
This section provides proofs of the theorems on multiplicities of eigenvalues 2, 4 stated in Section 1. In the first two subsections, we shall prove Theorems 1.3 and 1.4. As is seen below, a reason for large multiplicities of eigenvalues 2 or 4 of GC 2k,0 (X) is that the (2k, 0)-clusters also have large multiplicities of eigenvalues 2 or 4. On the other hand, it is considered that the structure of an initial graph X would affect the eigenvalue distribution of its Goldberg-Coxeter constructions. A few remarkable examples shall be provided in Section 4.3, where a proof of Theorem 1.5 is also included.
Proof. The function given in Figure 8 (a), where α ∈ R is arbitrary, is a D 3 -invariant eigenfunction with eigenvalue 4 for the (2, 0)-cluster. Also, the function given in Figure 8 (b) is a D 3 -invariant eigenfunction with eigenvalue 2 for the (4, 0)-cluster.   Proof of Lemma 4.2. We introduce the coordinate in (3.6) on the vertex set V(p) of the (2k, 0)cluster X(p). For each positive integer n, we set which is considered as the vertex set of an (n, 0)-cluster. The subgraph of the hexagonal lattice induced by V n is denoted by X n , which is identified with an (n, 0)-cluster. Also, for each a + bω, m + c + dω ∈ V n , we label for the corresponding equation of ∆ X(p) u = 4u as follows: where u is supposed to take the same value at a vertex outside V n as at the unique adjacent vertex of V n , such as E(0) : u(0) + u(m) + u(0) + u(0) = 0, E(1) : u(1) + u(m + 1) + u(m) + u(1) = 0. Let us first discuss the solvability of the following families of equations and the T 1 -invariance of the solutions, where T 1 : C → C is defined by T 1 (z) := ωz.
(1-a) {E(a + bω) | a + b = l, m ≤ a ≤ l − m}; assume that u is defined on and that u is invariant under T 1 on this set.
(1-b) {E(m + a + bω) | a + b = l, m ≤ a ≤ l − m}; assume that u is defined on and that u is invariant under T 1 on this set.
It is easily proved that (1-a) for any l and (1-b) for l odd are uniquely solvable and that each u of the solutions is invariant under T 1 on the set where u is newly defined. In the case where l is even, it is also proved from the T 1 -invariance of u that (1-b) is uniquely solvable if and only if u(l) = u(l + 1), and that the solution is invariant under T 1 . Let us define T 2 : C → C by T 2 (z) = T k 2 (z) := (1 − ω)z + (2k − 1)ω and T 3 : C → C by T 3 (z) = T k 3 (z) := −z + 2k − 1. We denote by (2-a) and (2-b) the families of equations transferred via T 3 from (1-a) and (1-b) respectively, and by (3-a) and (3-b) via T 2 . Then similar arguments as above (or simply symmetry of V 2k ) show the solvability and the T 2 -invariance (resp. T 3 -invariance) of the solutions of (2-a) and (2-b) (resp. (3-a) and (3-b)) provided u(2k −l−2+(l+1)ω) = u(2k −l−1+lω) (resp. u((2k − l − 1)ω) = u((2k − l − 2)ω)). We shall finish the proof by using the solvability and the symmetry of the solutions of (1-a)- (3-b) in an appropriate order.
Let us start by (i-b) with l = 2s − 1, m = 0 and with (4.2) for each i = 1, 2, 3. So far u is defined on with j = 0, on which u is invariant under the D 3 -action on V 4s+2 , and on which E is satisfied except on "the inside boundary": We assign α to the six large black vertices.
with j = 0. Now we assume that on (4.3) with j replaced by j − 1 ( j ≥ 1), u is defined and is invariant under D 3 -action on V 4s+2 and E is satisfied except on the inside boundary (4.4) with j replaced by j − 1. Then, by symmetry we can solve (i-a) with l = 2s − 1 + j, m = 2( j − 1) for i = 1, 2, 3, whose solution has the desired symmetry. It then follows the solvability and the symmetry of the solution of (i-b) with l = 2s − 1 + j, m = 2( j − 1) for i = 1, 2, 3. What we have to see is the solvability of (4.5) which is valid because of u(2a) = u(2a+1) and u(2aω) = u((2a+1)ω) for a = 0, 1, . . . , 2s. Now we conclude that u is defined on (4.3), where u is invariant under D 3 -action on V 4s+2 and E is satisfied except on the inside boundary (4.4).
A very similar proof works for eigenvalue 2 and the following is obtained. Lemma 4.3. A (2k, 0)-cluster has D 3 -invariant eigenvalue 2, whose multiplicity is exactly k/2 .
(1) The function given in Figure 11 (a), where α ∈ R is arbitrary, is a D 4 -invariant eigenfunction with eigenvalue 4.
(a) (4, 0)-cluster (b) (10, 0)-cluster Figure 11. D 4 -invariant eigenfunctions with eigenvalue 4 (2) Since the proof is again almost similar as that of Theorem 1.3, let us explain part of the differences. We introduce the same coordinate Z[i] as before on the vertex set V(p) of the (2k, 0)cluster X(p). For each positive integer n, we set The induced subgraph by V n is denoted by X n . Also, for each a + bi ∈ V n , we label for the corresponding equation of ∆ X(p) u = 4u as follows: E(a + bi) : u(a + 1 + bi) + u(a + (b + 1)i) + u(a − 1 + bi) + u(a + (b − 1)i) = 0, where u is supposed to take the same value at a vertex outside V n as at the unique adjacent vertex of V n . As in the proof of Theorem 1.3, we can construct a D 4 -invariant eigenfunction u on X 2k satisfying u(2a) = −u(2a + 1) by an inductive argument. Let us omit the remaining proof. (See Figure 11 (b) for an example. )