Cayley sum graphs and eigenvalues of (3, 6)-fullerenes

Matt DeVos  Luis Goddyn^{∗}  Bojan Mohar^{†}^{‡}
Robert Šámal^{§}^{¶}

Department of Mathematics Simon Fraser University Burnaby, B.C. V5A 1S6

                 email: {mdevos,goddyn,mohar,rsamal}@sfu.ca

Abstract We determine the spectra of cubic plane graphs whose faces have sizes 3 and 6. Such graphs, “(3,6)-fullerenes”, have been studied by chemists who are interested in their energy spectra. In particular we prove a conjecture of Fowler, which asserts that all their eigenvalues come in pairs of the form {λ, −λ} except for the four eigenvalues {3, −1, −1, −1}. We exhibit other families of graphs which are “spectrally nearly bipartite” in this sense. Our proof utilizes a geometric representation to recognize the algebraic structure of these graphs, which turn out to be examples of Cayley sum graphs.

Keywords: (3,6)-cage, fullerene, spectrum, Cayley sum graph, Cayley addition graph, geometric lattice, flat torus. MSC: 05C50, 05C25, 05C10

1 Introduction

A (3, 6)-fullerene is a cubic plane graph whose faces have sizes 3 and 6. These graphs have received recent attention from chemists due to their similarity to ordinary fullerenes. (Such graphs are sometimes called (3, 6)-cages in that community, but in graph theory this term already has a different, well-established meaning.) In 1995, Patrick Fowler (see [7]) conjectured the following result, which we prove here. Prior to this work, this result had been established for several subfamilies of (3, 6)-fullerenes [7, 5, 14]. Recall that the spectrum of a graph is the multiset of eigenvalues of its adjacency matrix.

Theorem 1.1. If G is a (3, 6)-fullerene, then the spectrum of G has the form {3, −1, −1, −1}∪L∪ −L where L is a multiset of nonnegative real numbers, and −L is the multiset of their negatives.

∗Supported by a Canada NSERC Discovery Grant

†Supported in part by ARRS Research Grant P1–0297, by an NSERC Discovery Grant, and by the CRC program.

‡On leave from: IMFM & FMF, Department of Mathematics, University of Ljubljana, Ljubljana, Slovenia.

§Supported by PIMS postdoctoral fellowship.

¶On leave from Institute for Theoretical Computer Science (ITI), Charles University, Prague, Czech Republic.

Figure 1: Examples of some small (0, 3, 6)-fullerenes.

In fact we prove an extended conjecture of Fowler et al. [7]. They propose that a generalized class of graphs called (0, 3, 6)-fullerenes also exhibit this “spectrally nearly bipartite” behavior. A semiedge of a graph is an edge with one endpoint, but unlike a loop, a semiedge contributes just one to both the valency of its endpoint and the corresponding diagonal entry of the adjacency matrix. In a plane embedding, a semiedge s with endpoint v is drawn as an arc with one end at v which sits in a face f, and s contributes one to the length of f. Figure 1 displays some examples of small (0, 3, 6)-fullerenes.

The outline of our proof is as follows. We show that every (0, 3, 6)-fullerene can be represented as a quotient of a certain lattice-like graph in the plane. This geometric description allows us to prove that these graphs are Cayley sum graphs. Then we call on a theorem which describes the spectral behavior of Cayley sum graphs in terms of the characters of the group.

In fact, the geometric description of (0, 3, 6)-fullerenes which is inherent in our proof is just a slight extension of a construction for (3, 6)-fullerenes which has been discovered by several authors [5, 7, 16], and follows easily from a deep theorem on the intrinsic metric of polygonal surfaces by Alexandrov [1]. In Section 4, we give a proper exposition of this construction, and a proof that it is universal.

With this construction in hand, it is possible to explicitly compute the spectrum of (0, 3, 6)- fullerenes, and in Section 5 we detail precisely how this computation can be made. Finally, in Section 6, we generalize to an arbitrary dimension, and expose the connection between this type of geometric graphs and Cayley sum graphs.

2 Cayley sum graphs

Let Γ be a finite additive abelian group, and let S ⊆ Γ. We define the Cayley sum graph CayS(Γ, S) to be the graph (V, E) with V = Γ, and uv ∈ E if and only if u + v ∈ S. If S is a multiset, then CayS(Γ, S) contains multiple edges, and if there exists u ∈ Γ with 2u ∈ S, then the edge uu is a semiedge. This definition is a variation of the well-studied Cayley graph Cay(Γ, S), in which uv forms an edge if and only if u − v ∈ S.

In contrast with Cayley graphs, there are only a few appearances of Cayley sum graphs in the literature (see [9] and references therein). For this reason we state some of their elementary properties. The graph G = CayS(Γ, S) is |S|-regular. While G is not generally vertex-transitive, the map x ↦ x + t is an isomorphism from G to CayS(Γ, S + 2t), for every t ∈ Γ. Finally, the squared graph G^{(2)}, which has an edge for each walk of length 2 in G, is the ordinary Cayley graph Cay(Γ, S − S) where S − S is the multiset {s_{1} − s_{2} | s_{1}, s_{2} ∈ S}.

The spectrum of a (finite abelian) Cayley graph Cay(Γ, S) is easy to describe (see [10, Ex. 11.8] or [11], where the nonabelian case is dealt with). Every character χ of Γ is a (complex-valued) eigenvector corresponding to the eigenvalue

χ(S) := ∑_{s∈S} χ(s) .

We may assume Γ = ℤ_{n_{1}} × ⋯ × ℤ_{n_{u}}, where |Γ| = ∏_{i} n_{i} and ℤ_{k} denotes the cyclic group of order k. To each a = (a_{1}, …, a_{u}) ∈ Γ we associate the group character

χ_{a} : (x_{1}, …, x_{u}) ↦ exp (2πi ∑_{j} \frac{a_{j}x_{j}}{n_{j}} ) .

The characters for a and −a satisfy χ_{−a}(x) = \overline{χ_{a}(x)}, so χ_{a} is a real-valued (indeed ±1-valued) eigenvector of Cay(Γ, S) if and only if a is an involutive group element. If a is not involutive, then the real and imaginary parts of χ_{a} provide real-valued eigenvectors for the conjugate pair of eigenvalues χ_{a}(S), χ_{−a}(S).

Cayley sum graphs exhibit a similar phenomenon. Let R = {χ_{a} | a + a = 0} be the real-valued characters of Γ, and let C be a set containing exactly one character from each conjugate pair \{χ_{a}, χ_{−a}\} (where a ∈ Γ and a + a ≠ 0). So the set of characters of Γ is R ∪ \{χ, \overline{χ} | χ ∈ C\}. Versions of the following result can be found in the literature [6, 2]. Theorem 2.1. Let G = CayS(Γ, S) be a Cayley sum graph on a finite abelian group Γ, and let R, C be as above. The multiset of eigenvalues of G is

{χ(S) : χ ∈ R} ∪ {±|χ(S)| : χ ∈ C}.

The corresponding eigenvectors are χ (for χ ∈ R), and the real and the imaginary parts of αχ (for χ ∈ C with a suitable complex scalar α which depends only on χ(S)). Proof. Let χ be a character of Γ and u ∈ Γ a vertex of CayS(Γ, S). Then

∑_{v∈N(u)} χ(v) = ∑_{s∈S} χ(s − u) = χ(S)\overline{χ(u)} .

This shows that every real-valued character is an eigenvector corresponding to the eigenvalue χ(S). If χ ∈ C, then χ is not an eigenvector. In this case we choose a complex number α such that |α| = 1 and α^{2}χ(S) = |χ(S)| and we define x(v) = αχ(v). It follows that for every u ∈ Γ,

∑_{v∈N(u)} x(v) = α^{2}χ(S) · α^{−1}\overline{χ(u)} = |χ(S)| · \overline{x(u)}.

Consequently, Re x and Im x are real eigenvectors corresponding to eigenvalues |χ(S)| and −|χ(S)|, respectively. Both of these vectors are nonzero, as they generate the same 2-dimensional (complex) vector space as \{χ, \overline{χ}\}. This, together with the orthogonality of characters, implies that we have described the complete set of eigenvectors, and thus the entire spectrum of CayS(Γ, S).

3 (0, 3, 6)-fullerenes as Cayley sum graphs

The goal of this section is to prove that (0, 3, 6)-fullerenes are Cayley sum graphs, and to subsequently prove Fowler’s conjecture regarding their spectra.

The proof of Theorem 3.1 utilizes structural properties of 3-regular hexangulations of the torus. This class of graphs was classified by Altshuler [4] and studied by many others (e.g., Thomassen [15]). In a recent work of Alspach and Dean [3], it is shown that they are indeed Cayley graphs, and a description of the group is given. Although the properties we require of these graphs are similar to those found elsewhere, our approach is novel since it is inherently geometric.

A polygonal surface H is a connected 2-manifold without boundary which is obtained from a collection of disjoint simple polygons in 𝔼^{2} by identifying them along edges of equal length. Thus we view H both (combinatorially) as an embedded graph with vertices, edges, and faces, and as a manifold with a (local) metric inherited from 𝔼^{2}.

Theorem 3.1. Every (0, 3, 6)-fullerene is isomorphic to a Cayley sum graph for an abelian group which can be generated by two elements.

Proof. Let G be a cubic (0, 3, 6)-fullerene with vertex set V. Let G_{2} = G × K_{2} (the categorical graph product). Let (V_{•}, V_{∘}) be the corresponding bipartition of V (G_{2}), and for every v ∈ V, let v_{•} ∈ V_{•} and v_{∘} ∈ V_{∘} be the vertices of G_{2} which cover v. Every semiedge vv ∈ E(G) lifts to the edge v_{•}v_{∘} in G_{2}. Each facial walk of G bounding a face of size 6 lifts to two closed walks of length 6 in G_{2}, and each facial walk of G bounding a face of size 3 lifts to a closed walk of length 6 in G_{2}. Accordingly, we may extend G_{2} to a polygonal surface H by treating all edges as having equal length and adding a regular hexagon to each closed walk which is the pre-image of a facial walk of G, with clockwise orientation as given by the clockwise orientation of that face. Now, H is an orientable polygonal surface, all vertices have degree three, and all faces are regular hexagons, so H is a regular hexangulation of the flat torus. Let H̃ be the universal cover of H and let p : H̃ → H be the covering map. Then H̃ (with the metric inherited from H) is the regular hexangulation of the Euclidean plane. We define Ṽ_{•} = p^{−1}(V_{•}), Ṽ_{∘} = p^{−1}(V_{∘}), and x̃ = p^{−1}(x) for x ∈ V_{•} ∪ V_{∘}.

Fix a vertex u_{•} ∈ V_{•}, and treat H̃ as a regular hexangulation of 𝔼^{2} with p((0, 0)) = u_{•}. This equips H̃ with an (additive abelian) group structure. The point set Ṽ_{•} is a geometric lattice. The point set ũ_{•} is a sublattice of Ṽ_{•}. Any fundamental parallelogram of ũ_{•} is a fundamental region of the cover p. We may identify H with H̃/ũ_{•}, and this equips H with a group structure whose identity is u_{•}.

For every y ∈ H (y ∈ H̃) the map x ↦ x + y is an isometry of H (H̃, respectively). This map may or may not preserve the combinatorial structure of H (H̃). An isometry μ : H → H is respectful if μ is an automorphism of the embedded graph associated with H. An isometry μ̃ : H̃ → H̃ is respectful if it is a lift of a respectful isometry of H. Now, for every y ∈ Ṽ_{•} the map x ↦ x + y is a respectful isometry of H̃. Accordingly, V_{•} is a subgroup of H with identity u_{•}, and for every y ∈ V_{•} the map x ↦ x + y is a respectful isometry of H.

Let ρ be the automorphism of the graph G_{2} given by the rule ρ(v_{∘}) = v_{•} and ρ(v_{•}) = v_{∘} for every v ∈ V. Now, ρ extends naturally to a respectful isometry of H which preserves the orientation of the hexagons, but interchanges V_{∘} and V_{•}. We choose a respectful isometry ρ̃ of 𝔼^{2} so that ρ lifts to ρ̃. Because ρ̃ preserves the orientation of 𝔼^{2}, the isometry ρ̃ is either a rotation or translation.

Since ρ̃ is respectful and maps Ṽ_{•} to Ṽ_{∘}, it easily follows that ρ is either a rotation by π about the center of an edge or face, or ρ is a rotation by π/3 about the center of a face F.

We first consider the latter case. Here, all three vertices of Ṽ_{•} which are on the boundary of F, lie in the same orbit of ρ̃^{2}. Since ρ^{2} is the identity, all three vertices cover the same vertex, say v_{•} in H. The other three vertices of F cover v_{∘}. In this case G_{2} is the theta-graph with vertex set {v_{•}, v_{∘}}. Here we have G ≅ CayS({0}, {0, 0, 0}), the graph with one vertex and three semiedges, and there is nothing left to prove.

We henceforth assume that ρ̃ is a rotation by π. Let x, y ∈ V_{•} and choose x̃, ỹ ∈ Ṽ_{•} which project (respectively) to x, y. Then (using the fact that ρ̃ is a rotation by π) we find that

ρ(ρ(x) + y) = p(ρ̃(ρ̃(x̃) + ỹ))
= p(x̃ − ỹ)
= x − y.

In other words, for any fixed y ∈ V_{•}, conjugating the map on H given by x ↦ x + y, by ρ yields the map x ↦ x − y.

We define a labeling ℓ : V_{•} ∪ V_{∘} → V_{•} by the rule ℓ(v_{•}) = ℓ(v_{∘}) = v_{•}. We regard ℓ to be a labeling of V (G_{2}) by elements of the abelian group V_{•}. Let v ∈ V and let y ∈ V_{•}. Then we have

ℓ(v_{•} + y) = ℓ(v_{•}) + y

and

ℓ(v_{∘} + y) = ℓ(ρ(v_{∘} + y))
= ℓ(ρ(ρ(v_{•}) + y))
= ℓ(v_{•} − y)
= ℓ(v_{∘}) − y.

That is, the group V_{•} acts on the labels of points in V_{•} by addition and on the labels of points in V_{∘} by subtraction. Let S be the multiset of labels of the three vertices in V_{∘} which are adjacent to u_{•} (recall that u_{•} is the group identity for V_{•}). Then, for every v_{•} ∈ V_{•}, the three neighbors of v_{•} in G_{2} have labels S − v_{•}. In particular, v and v′ are adjacent vertices in G if and only if ℓ(v_{•}) + ℓ(v′_{∘}) ∈ S. It follows immediately from this that G ≅ CayS(V_{•}, S). Since Ṽ_{•} can be generated by two elements, V_{•} = Ṽ_{•}/ũ_{•} can also be generated by two elements, and this completes the proof.

We need only one quick observation before we resolve Theorem 1.1 and the extended conjecture of Fowler et al. If G is a cubic plane graph with s semiedges, and f_{i} faces of size i for every i ≥ 1, then 3|V (G)| = 2|E(G)| − s = ∑_{i≥1} if_{i}. Applying Euler’s formula, we find that ∑_{i≥1}(6 − i)f_{i} = 12 − 3s. In particular, every (0, 3, 6)-fullerene satisfies

s + f_{3} = 4.  (1)

Theorem 3.2. If G is a (0, 3, 6)-fullerene with s semiedges, then the spectrum of G may be partitioned as {M, L, −L} where one of the following holds:

(a) s = 0 and M = {3, −1, −1, −1},

(b) s = 2 and M = {3, −1},

(c) s = 3 and M = {3}, or

(d) s = 4 and M = {3, 1}.

Proof. By the previous theorem, there is an abelian group Γ which can be generated by two elements so that G ≅ CayS(Γ, S) for some S ⊆ Γ with |S| = 3. By Theorem 2.1, we may partition the eigenvalues of G into multisets M, L, −L where M = {χ(S) : χ ∈ R} and R is the set of ±1-valued characters of Γ. Every eigenvalue in M is the sum of three integers in {±1}. The identity character corresponds to 3 ∈ M. Since G is not bipartite, we have −3 ∉ M, so every other element of M is ±1. The trace of the adjacency matrix is equal to s, and is also equal to the sum of the eigenvalues. Since L and −L sum to 0, we conclude that s = ∑ M.

We have |R| ∈ {1, 2, 4} because Γ has 2^{k} involutive elements, for some k ≤ 2. If |R| = 1, then M = {3} and s = 3 as in the statement. If |R| = 2, then s = ∑ M = 3 ± 1, so we have either the case s = 2 or s = 4 of the statement. Finally, we assume |R| = 4. By Equation (1) we have s ≤ 4, so ∑ M ∈ {0, 2, 4}. If ∑ M = 0, then s = 0 (G is a (3, 6)-fullerene), and we have case (a). Finally, if ∑ M ∈ {2, 4}, then M contains both a 1 and a −1. By transferring these two entries from M to the multisets L and −L, we find ourselves again in either the case s = 2 or the case s = 4 of the statement. This completes the proof.

We remark that there are infinitely many (0, 3, 6)-fullerenes with s semiedges, for each s = 0, 2, 3, 4. As shown by Theorem 3.2, there are none with s = 1, a fact that is non-trivial to prove from the first principles (compare Theorem 2 (with k = 3) in [8, Sec. 13.4, p. 272]).

4 An explicit construction

It is known that all (3, 6)-fullerenes arise from the so-called grid construction. Roughly speaking, the grid construction expresses the dual plane graph, which is a triangulation of the sphere, as a quotient of the regular triangular grid. The grid construction is also used by physicists [5, 14] (sometimes without formal justification) since it a convenient way to classify (3, 6)-fullerenes and compute their invariants.

We describe an extension of the grid construction and show that it characterizes the (0, 3, 6)- fullerenes. The construction makes clear how semiedges arise. The group structure of (0, 3, 6)- fullerenes is explicitly determined as a quotient of the A_{2} lattice group. With this, we can easily find the Cayley sum graph representation via standard lattice computations, and thereby determine the spectrum and the eigenvectors of every (0, 3, 6)-fullerene.

In the following, T is the infinite triangular grid, whose vertices (called gridpoints) are points in the A_{2} lattice. The midpoint of any edge in T is called an edgepoint. The dual G^{∗} of a plane graph G with semiedges is defined as an obvious extension of the dual of an ordinary graph; every semiedge in G which is incident with vertex v and face f corresponds to a semiedge in G^{∗} which is incident with the dual vertex f^{∗} and the dual face v^{∗}.

Construction 4.1. The following procedure results in a (0, 3, 6)-fullerene, G.

Figure 2: An example of Construction 4.1.

1. Let △ABC be a triangle having no obtuse angle, and whose vertices are gridpoints of T . Let Ā, B̄, C̄ be the midpoints of the edges which are opposite to A, B, C (respectively) in △ABC.

2. Optionally, translate △ABC so that A coincides with an edgepoint of T .

3. From △ABC, we fold an (isosceles) tetrahedron Q = AĀB̄C̄ by identifying the boundary segment ĀB with ĀC, B̄C with B̄A, and C̄A with C̄B (so A, B, and C are identified into a single vertex in Q). The portion of T lying within △ABC becomes a finite graph G^{∗}, possibly with semiedges, and drawn on the surface of Q.

4. Let G be the dual of the plane graph G^{∗}.

Every gridpoint within or on the boundary of △ABC, except A, Ā, B̄, and C̄, has degree 6 in G^{∗}, and corresponds to a hexagonal face of G. After Step 2, each of A, Ā, B̄, C̄ is either a gridpoint or an edgepoint of T . If X ∈ {A, Ā, B̄, C̄} is a gridpoint, then X becomes a vertex of degree 3 in G^{∗}, and corresponds to a triangular face in G. If X is an edgepoint, then X becomes one end of a semiedge in G^{∗}, which corresponds to a semiedge in G. It follows that Construction 4.1 results in a (0, 3, 6)-fullerene.

We remark that Construction 4.1 works even if △ABC has an obtuse angle (although it does not yield a geometric tetrahedron). However, this does not give any new (0, 3, 6)-fullerenes, as the following theorem shows. By forbidding obtuse triangles, we lose no generality and gain canonicality.

Theorem 4.2. Every (0, 3, 6)-fullerene arises from Construction 4.1.

Proof. Let G be a (0, 3, 6)-fullerene. If G has just one vertex, then G arises from the construction when △ABC is a triangular face of T . We assume G has at least two vertices. The proof of Theorem 3.1 shows that the direct product G_{2} = G × K_{2} is a bipartite hexangulation H of the flat torus, where H is the image of a covering map p : H̃ → H from a hexagonal tessellation of the plane.

We further recall that there is an isometry ρ of H which is respectful of G_{2} and interchanges its partite sets V_{•} and V_{∘}. This isometry lifts to an isometry of H̃ which is a rotation ρ̃ by π about a point, say A ∈ H̃, which is either the center of a hexagonal face, or the midpoint of an edge of H̃. (More precisely ρ̃ : x ↦ 2A − x is the central symmetry through A.) The kernel of p is a geometric lattice Λ in H̃, and rotation by π about any point in in the scaled lattice \frac{1}{2}Λ also projects to ρ. Let B, C be points in H̃ such that the vectors AB, AC form lattice basis for Λ. By possibly translating C by a (unique) integer multiple of AB, we can assume that △ABC has no obtuse angles. This lattice basis defines a fundamental parallelogram ABDC where AD = AB + AC. Scaling the parallelogram by \frac{1}{2} results in a fundamental parallelogram for \frac{1}{2}Λ whose vertices we may label AC̄ĀB̄ as in Construction 4.1.

Now each vertex v of G lifts to a unique pair of vertices v_{•}, v_{∘} in the (half-open) parallelogram ABDC, where v_{•}, v_{∘} are centrally symmetric about Ā. By representing v with the vertex in {v_{•}, v_{∘}} which lies in △ABC, and identifying boundary segments of △ABC as in Step 3 of the construction, we obtain an isomorphic copy of G. Finally, Construction 4.1 is stated in terms of the triangular grid T , which is the plane dual of H̃.

We remark that Construction 4.1 in fact produces a (0, 3, 6)-fullerene G rooted at a triangle or semiedge labeled with A. Two triangles drawn in T result in isomorphic pairs (G, A) if and only if the triangles are congruent. Therefore the map △ABC ↦ G is at most 4-to-1 up to symmetries of T .

5 Computing the spectrum

In this section, we use Construction 4.1 to compute the group and spectrum of any particular (0, 3, 6)-fullerene G.

The faces of T consist of up-triangles (Δ) and down-triangles (∇). We regard Λ_{•} to be an A_{2}-lattice generated by unit-length vectors a, b with ∠ab = π/3. With A being the gridpoint selected in Step 1 of Construction 4.1, we shall assume that the origin of Λ_{•} is (the center of) the up-triangle u_{•} := △A(A + a)(A + b). Note that Λ_{•} is a translation of the gridpoints of T and corresponds to Ṽ_{•} in the proof of Theorem 3.1. We denote by Λ the sublattice of Λ_{•} generated by vectors \overrightarrow{AB} and \overrightarrow{AC}. (A translation of Λ is used in the proof of Theorem 4.2.) In Step 2, we translate △ABC by a vector

c := \frac{p_{1}}{2} a + \frac{p_{2}}{2} b  (2)

for integers p_{1}, p_{2}. We may assume without loss of generality that p_{1}, p_{2} ∈ {0, 1}, so, after Step 2, the point A is either a vertex or an edgepoint on the boundary of u_{•}. Let p, q, r, s be integers satisfying

AB = pa + qb,  AC = ra + sb.  (3)

Let Ā, B̄, C̄ and T be as in the construction of G.

To express G as a Cayley sum graph we label the faces of T with elements of the finite abelian group presented as Γ = ⟨ α, β | pα + qβ = 0, rα + sβ = 0 ⟩. We define f : Λ_{•} → Γ by

f(ia + jb) = iα + jβ,  (4)

and extend f to the down-triangles in such a way that triangles which are centrally symmetric with respect to A receive the same value of f. The kernel of f is (a translation of) the lattice Λ generated by AB and AC. We observe the following properties:

• f assigns the same value to triangles that are identified during the ‘folding’ stage of the construction. This is because the triangles that are identified are symmetric with respect to each of C̄, B̄, and Ā; each of these symmetries is a composition of the negation map Λ_{•} ↦ −Λ_{•} and a translation by an element of Λ = ker f.

• f is a bijection from V (G) to Γ. By construction, the up-triangles within the fundamental region ABDC correspond to elements of Γ. The down-triangles within the triangle ABC correspond to up-triangles within DCB.

• If u_{1} and u_{2} are two up-triangles, then f(u_{2}) = f(u_{1}) + f(u_{2} − u_{1}). If d_{1} and d_{2} are two down-triangles then f(d_{2}) = f(d_{1}) − f(d_{2} − d_{1}).

Now let u be any up-triangle and d_{1}, d_{2}, d_{3} its neighbors. We define the sum-set S = {f(u)+f(d_{i}) | i = 1, 2, 3}. From the above-mentioned properties of f it follows that S does not depend on the choice of u. The symmetry around A shows that we get the same sum-set if we consider neighbors of a down-triangle to define S. It follows that G ≅ CayS(Γ, S).

We can explicitly compute Γ and S by applying standard lattice computations. We recall that the Smith normal form of a nonsingular integer matrix M is the unique matrix diag(δ_{1}, δ_{2}, …, δ_{k}) = UMV where U and V are unimodular and the product δ_{1}δ_{2} ⋯ δ_{i} is the g.c.d. of the order i subdeterminants of M, 1 ≤ i ≤ k (see, e.g. [12, Section 4.4]).

Lemma 5.1. Let G be a (0, 3, 6)-fullerene obtained from Construction 4.1, and let c, p, q, r, s be

as in (2) and (3). Let diag(m, n) = UMV be the Smith normal form of the matrix M = (\begin{matrix}p & r \\ q & s \end{matrix}).

Let u, v denote the columns of U. Then G = CayS(Γ, S) where Γ = ℤ_{m} × ℤ_{n} and

S = { (p_{1} − 1)u + p_{2}v, p_{1}u + (p_{2} − 1)v, (p_{1} − 1)u + (p_{2} − 1)v }.

Here we interpret each column vector \binom{x_{1}}{x_{2}} ∈ S to be the group element (x_{1} mod m, x_{2} mod n) ∈ Γ.

Proof. The columns of the matrix B := (a, b) form a lattice basis for Λ_{•} whereas those of BM generate the sublattice Λ. Since U and V are unimodular, the columns of B′ := BU^{−1} also generate Λ_{•}. Accordingly, Λ is generated by the columns of BMV = B′ diag(m, n). It follows that Γ = Λ_{•}/Λ ≅ ℤ_{m} × ℤ_{n}. If we index the up-triangles with respect to the basis B′, then the mapping f : B′\binom{i′}{j′} ↦ (i′ mod m, j′ mod n) is the one defined in (4). Changing the basis to B = B′U, we find f(ia + jb) = iu + jv.

After Step 1 of the construction, the three down-triangles which are neighbours of u_{•} reflect through A to the up-triangles at −a, −b and −a − b. When A is translated by c in Step 2, the three up-triangles are accordingly translated by 2c = p_{1}a + p_{2}b. Therefore

S = {f((p_{1} − 1)a + p_{2}b), f(p_{1}a + (p_{2} − 1)b), f((p_{1} − 1)a + (p_{2} − 1)b)}

as claimed.

We present a sample computation illustrating the determination of the group and spectrum.

Example 5.2. The example of Figure 2 corresponds to (p_{1}, p_{2}) = (0, 0) and (p, q, r, s) = (6, 2, −2, 6). All six integers are even, so the resulting graph G has no semiedges. We compute the Smith normal form to be

UMV = (\begin{matrix}0 & 1 \\ −1 & −7 \end{matrix}) (\begin{matrix}6 & −2 \\ 2 & 6 \end{matrix}) (\begin{matrix}−2 & −3 \\ 1 & 1 \end{matrix}) = (\begin{matrix}2 & 0 \\ 0 & 20 \end{matrix}) .

Hence Γ = ℤ_{2} × ℤ_{20}. Furthermore, the generating set is

S = {−u + 0v, 0u − v, −u − v} = {(0, 1), (−1, 7), (−1, 8)}.

This implies G has eigenvalues 3, −1, −1, −1, and

{±|ε^{b} + (−1)^{a}ε^{7b} + (−1)^{a}ε^{8b}| : 0 ≤ a ≤ 1, 1 ≤ b ≤ 9} ,

where ε = e^{2πi/20}.

If we were to translate △ABC by (\frac{1}{2}a, 0), then we get a (0, 3, 6)-fullerene G′ with four semiedges. Here we have (p_{1}, p_{2}) = (1, 0), which has the effect of translating the generating set by u. That is,

G′ = CayS( ℤ_{2} × ℤ_{20}, {(0, 0), (1, 6), (1, 7)}), and
spec(G′) = {3, 1, 1, −1} ∪ {±|1 + (−1)^{a}ε^{6b} + (−1)^{a}ε^{7b}| : 0 ≤ a ≤ 1, 1 ≤ b ≤ 9} .

The four semiedges are incident with the vertices (0, 0), (1, 0), (0, 10), (1, 10) ∈ Γ.

6 The geometry of Cayley sum graphs

In Section 4 we saw how the geometric description of (0, 3, 6)-fullerenes in terms of the A_{2} lattice implies that they are Cayley sum graphs. Therefore their eigenvectors are easy to calculate, and their spectra are “nearly bipartite.” Here we describe the precise circumstances under which Cayley sum graphs arise from geometric lattices in this manner. In fact we will see that every Cayley sum graph arises as a quotient of two cosets of a geometric lattice. We then exhibit some families of Cayley sum graphs which have a recognizable crystallographic local structure.

First we review, in greater generality, the conditions under which a graph G is a Cayley sum graph. Let G_{2} be the cover G × K_{2} with bipartition (V_{•}, V_{∘}). Note that G_{2} has a natural automorphism, ρ, – we call it the inversion map – which transposes the two vertices within each fibre. By following the proof of Theorem 1.1, we find that G is a Cayley sum graph on an abelian group Γ if and only if Γ acts regularly on each of V_{•} and V_{∘} as a group of G_{2}-automorphisms, and this action satisfies

ρ^{−1}gρ = −g, for each g ∈ Γ.  (5)

Our construction proceeds with a sequence of graphs

G̃ ↦ G̃_{2} ↦ G_{2} ↦ G.

Let Λ_{•} ⊂ 𝔼^{d} be a geometric lattice, and let G̃ = CayS(Λ_{•}, S) be a Cayley sum graph with edges drawn as straight line segments. Each generator s ∈ S corresponds to a set of edges of G̃ whose midpoints are concurrent at the point \frac{1}{2}s. Let Λ_{∘} be any nontrivial coset of Λ_{•}, and let A ∈ ℝ^{d} be such that Λ_{∘} = 2A + Λ_{•}. Let ρ̃ : x ↦ 2A − x be the inversion map through A. As above, we construct G̃_{2} = G̃ × K_{2} with partite sets (Ṽ_{•}, Ṽ_{∘}) = (Λ_{•}, Λ_{∘}), where the fibres of G̃_{2} are the orbits of ρ̃.

The graph G̃_{2} is drawn in Euclidean d-space 𝔼^{d} with straight line segments for edges. Let 𝔼^{d}/ρ̃ denote the quotient space (an orbifold) whose points are the ρ̃-orbits {x, ρ̃(x)}, x ∈ 𝔼^{d}. Geometrically speaking, 𝔼^{d}/ρ̃ is a cone with its apex A having the solid angle of a halfspace. By mapping each point in 𝔼^{d} to its ρ̃-orbit, we may view G̃ ≅ G̃_{2}/ρ̃ as being naturally embedded in 𝔼^{d}/ρ̃. Every edge of G̃_{2} whose midpoint is A folds to a semiedge of G̃. In the case of (0, 3, 6)- fullerenes, G̃_{2} is the plane hexagonal grid, and G̃ is a grid drawn on a cone where every face is a hexagon except at A, where A is either the midpoint of a triangular face, or the end of a semiedge.

Now let Λ be any sublattice of Λ_{•}, and let p be the be the natural projection from 𝔼^{d} to the d-torus 𝔼^{d}/Λ. Then G_{2} := p(G̃_{2}) is a finite bipartite graph with partite sets (V_{•}, V_{∘}) := (p(Λ_{•}), p(Λ_{∘})), and which is embedded in 𝔼^{d}/Λ. Then ρ̃ projects to ρ, a symmetry of order 2 in the d-torus. Evidently ρ is an inversion map for G_{2} satisfying (5) with Γ = Λ_{•}/Λ. Therefore G ≅ G_{2}/ρ is a finite Cayley sum graph embedded in the orbifold 𝔼^{d}/ρΛ. Let A ⊂ 𝔼^{d}/Λ be the fixed points of ρ. Then A consists of exactly 2^{d} points having the form p(A + \frac{1}{2}Λ) and ρ acts on 𝔼^{d}/Λ as an inversion through any point in A. As an orbifold, 𝔼^{d}/ρΛ is orientable if and only if d is even. To visualize 𝔼^{d}/ρΛ, it is convenient to select a fundamental region for 𝔼^{d}/Λ whose 2^{d} extreme points belong to A + Λ. Let T be the part of the region which lies on the positive side of a hyperplane H, which contains the region’s centroid. All points in A lie on the boundary of T so we obtain 𝔼^{d}/ρΛ by an appropriate gluing of the boundary of T. The graph G is embedded in T with each vertex {x, ρ(x)} represented by the unique point in {x, ρ(x)} ∩ T. For example, 𝔼^{2}/ρΛ is an isosceles tetrahedron, whose four extreme points comprise A. The grid construction of (0, 3, 6)-fullerenes corresponds to selecting H to be a diagonal of a fundamental parallelogram. The Cayley sum graph G has one semiedge for every point of A which lies on an edge of G_{2}. Figure 3 summarizes the commuting projections and the four embedded graphs.

Since every finite abelian group is the quotient of two geometric lattices, it follows that every finite Cayley sum graph G arises from a quadruple (Λ_{•}, S, A, Λ) as described above. By employing a linear transformation we can even assume that Λ_{•} = ℤ^{d}. We do not make this assumption here, since that would obfuscate the following examples. When the sum set S is a set of lattice points which are close to 2A, then each edge of G̃_{2} is a short line segment, and G̃_{2} is often a recognizable bipartite crystallographic configuration. After selecting Λ and applying the above construction, we obtain a finite Cayley sum graph embedded in T with a local geometry that reflects the crystallographic structure of G̃_{2}. We present with some examples.

• For d = 1, if G̃_{2} is the two-way infinite path, then G̃ is the infinite ray with a semiedge at its origin, and G_{2} is an even cycle drawn as a regular polygon. The inversion ρ identifies points reflected in a line which bisects opposite edges of the polygon, so G is a finite path with a semiedge at each end. The spectrum of G takes the form M ∪ L ∪ −L where M = {2} or M = {2, 0}.

• (Grid-like examples) If Λ_{•} = D_{d}, the lattice of integer points of even weight, and Λ_{∘} = Λ_{•} + (1, 0, 0, …), then Λ_{•} ∪ Λ_{∘} = ℤ^{d}, and we may take G̃_{2} to be the standard cartesian grid. If A = (\frac{1}{2}, 0, 0, …), then applying the construction with any sublattice Λ of Λ_{•} leads to a Figure 3: Constructing finite Cayley sum graph from a lattice. Illustrated with the D_{2}-lattice, resulting in a 28-vertex Cayley sum graph which is also a 4-regular quadrangulation of the tetrahedron.

Cayley sum graph G having exactly 2^{d} semiedges. If d = 2, then G is a 4-regular quadrangulation of an isosceles tetrahedron, with a semiedge at each tetrahedral vertex. Such an graph illustrates Figure 3. The set of unmatched eigenvalues of G is either M = {4} or M = {4, 0}. Indeed, every 4-regular quadrangulation of a sphere can be expressed in this way. To see this fact, we need only adapt the proof of Theorem 3.1. When d is odd, we may take A = (\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, …). Since A is not on an edge of the cubic grid, this results in a grid-like Cayley sum graph G having fewer than 2^{d} semiedges. Indeed G has no semiedges at all if Λ is a sublattice of 2Λ_{•}.

• (Diamond-like examples) Again we take Λ_{•} to be the D_{d}-lattice, but put Λ_{∘} = Λ_{•}+(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, …). The set Λ_{•} ∪ Λ_{∘} is commonly called the generalized diamond packing, and is denoted by D_{d}^{+} (see [13, p. 119]). The diamond grid is the graph G̃_{2} in which each point in Λ_{•} is joined to the 2^{d−1} nearest points in Λ_{∘}. Putting A = (\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, …) results in a Cayley sum graph having at least 2^{d−1} semiedges. A more attractive option is to put A = (\frac{5}{4}, \frac{1}{4}, \frac{1}{4}, …), which lies on no edge of G̃_{2}. Provided that Λ is a sublattice of 2Λ_{•}, this results in a Cayley sum graph having no semiedges. When d = 3, this construction gives a class of Cayley sum graphs having the local structure of diamond crystal. Such graphs satisfy M = {4, −2, −2} or M = {4, 0, −2, −2}. Another attractive class is based on D_{8}^{+}, otherwise known as the E_{8} lattice.

• The 24-dimensional Leech lattice Λ_{24} arises as the union of two cosets of a lattice hΛ_{24} which is obtained from the binary Golay code (see [13, p. 124]). This yields a particularly attractive class of crystallographic Cayley sum graphs of high dimension.

We have constructed infinite families of Cayley sum graphs whose spectra have the form M ∪ L ∪ −L, where M is a fixed finite multiset. By taking the categorical products with a fixed graph H, one obtains other “spectrally nearly bipartite” families of graphs. It would be interesting to find other natural examples of this phenomenon.

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