Equilateral Spherical Drawings of Planar Cayley Graphs

In this paper, we study equilateral spherical drawings of planar Cayley graphs. We focus on the case when the underlying group is generated by two rotations. In this case, the set of equilateral drawings can be parameterized by spherical ellipses on the unit sphere. Besides, we give an explicit formula to describe the shortest equilateral spherical drawing and the longest spherical equilateral drawing. Furthermore, we studied the drawing of Schreier coset graphs arising from these equilateral drawings.


Introduction
Cayley graph is a graph encoding the structure of a group, which is a central tool in combinatorial and geometric group theory. Given a group G with a symmetric generating set S, the Cayley graph Cay(G, S) associated to (G, S) is an undirected graph with the vertex set G such that two vertices x and y are adjacent if x = ys for some s ∈ S. The group G acts canonically on the graph Cay(G, S) by left multiplication and the action is transitive on the set of vertices.
A graph is planar if it has an embedding on the plane without edge-crossing. When all the edges of the embedding have the same length, it is called an equilateral embedding. Eades and Warmold [5] showed that determining whether a 2-connected planar graph has an equilateral embedding is NP-hard. Markenzon and Paciornik [13] presented a linear time algorithm to determine whether a 2-connected chordal graph has a equilateral embedding without edge-crossing.
One important subject in graph drawing is to find a nice drawing algorithm in 3D. For trees, there are several 3D graph drawing algorithms like [1], [9], [12]. For other particular kinds of graphs, there are also some 3D graph drawing algorithms like [6], [11], [3] and [4].
Maschke [14] classified planar Cayley graphs in 1896. A complete list of planar Cayley graphs can be also be found in [7]. He showed that when X = Cay(G, S) is a planar Cayley graph, G is isomorphic to a group of isometries of the unit sphere S 2 in R 3 . In other words, one can draw X in R 3 such that the vertices lie in S 2 , the edges are straight lines, and the action of G on X can be extended to S 2 as isometries. We call such a drawing a symmetric spherical drawing. However, Maschke's proof does not give an explicit construction of symmetric spherical drawings. He only drew the figures of all planar Cayley graphs and identify them as the skeletons of uniform polyhedrons.
To construct a symmetric spherical drawing explicitly, first we need find the way to identify G as a group of isometries of S 2 . This is equivalent to find a particular three dimensional real representation of G such that the image of G is the symmetric group of the corresponding uniform polyhedron. It is not hard to find such a representation by cases, but in fact there is a unified method using the method of spectral drawing [8,10], which will be introduced in Section 2.4.
After identifying G as a group of isometries on S 2 , one can fix a unit vector u as the vector used to draw the identity element of G. Then the coordinate of the vertex g is given by g u. Denote the resulted drawing by X u . Note that all X u 's are highly symmetric, since they all admit G as a group of isometries. When all edges of X u are of equal length, X u is called an equilateral spherical drawing. In this paper, we will like to study all equilateral spherical drawings via studying the following set: For example, let G = A 5 , the alternating group of degree 5, and S = {(12)(34), (12345), (15432)}. The Cayley graph X = Cay(G, S) is the truncated icoshedral graph. One can identify G as a group of isometries on S 2 from the spectral drawing. (In this case, G is also known as the icosahedral group. See [2] for more details.) The Figure 1 demonstrates X u for some random choices of u. The Figure 2 shows the set D, which is a union of two so-called spherical ellipses.
In the set D, the drawing with shortest edge length is indeed the skeleton of a truncated icosahedron; the drawing with longest edge length is the skeleton of an icosahedron. Besides, there exists some drawing equal to the skeleton of a dodecahedron as shown in Figure 3.
Note that the skeleton of a dodecahedron and the skeleton of an icosahedron are the drawings of the Schreier coset graphs X H associated to (A 5 /H, HSH/H) where H = ρ((123)) and ρ((12534)) , respectively. (See Section 2.1 for the definition of Schreier coset graph.) In general, we would like to know when a Schreier coset graph can be seen by certain X u .
In this paper, we will only focus on the case that G only consists of rotations as a group of isometries of S 2 . In this case, we call Cay(G, S) a rotational planar graph. The classification of such graphs is given in Section 2.3. The paper is organized as follows. In Section 2, we recall some background knowledge. In Section 3, we characterize the set D and give an explicit form of u for the shortest and longest equilateral spherical drawings of X u . Besides, we study the isomorphism class of X u . Especially, we show that the shortest and longest drawings are both unique up to isomorphism. Furthermore, we find some particular subgroups H such that there exists some X u which is also a graph drawing of the Schreier coset graph X H . In Section 4, we list the result for all rotational planar graphs.

Schreier coset graphs
For a subgroup H of G, define which is the H-double coset containing S. It can be also written as a disjoint union of left H-coset, denoted by HSH/H.
The Schreier coset graph X H associated to (G/H, HSH/H) is the graph in which its vertices are left H-cosets {gH|g ∈ G} in G and two vertices g 1 H and g 2 H are adjacent if g 2 H = g 1s H for somesH ∈ HSH/H.
It is well-known that a vertex-transitively graph may not be a Cayley graph, but it is always a Schreier coset graph [15].

Displacement function
Let R = R(θ, u) be a linear rotation around the unit vector u in R 3 of degree θ. The square displacement function of a rotation R on R 3 is defined by Denote by d R,max the maximum value of d R ( x) on the unit sphere S 2 .
(Shortest length drawing) (Longest length drawing) Figure 3: Some drawings X u with u ∈ D.

Classification of Rotational Planar Groups
A group G is called a planar group if there exists some generating set S such that Cay(G, S) is a planar graph. Maschke showed that if G can be identified as a group of isometries of S 2 . In this case, G contains rotations and reflections. If G only contains rotations, G is a rotation group and its classification can be also found in [7, Theorem 6.3.1]. The following theorem is the result of Maschke's work [14].

Spectral Drawings
Let X be a finite connected undirected graph with the vertex set V = {v 1 , · · · , v n } and the edge set E. Let R[V ] be a real inner product space with an orthonormal basis α = { e v1 , · · · , e vn }. The Laplacian operator L is a linear transformation on R[V ] characterized by The Laplacian operator is positive semi-definite with eigenvalues , · · · , u n be an orthonormal eigenbasis such that L( u i ) = λ i u i for all i.
Then the k-dimensional spectral drawing is a straight-line drawing of X onto R k such that the coordinate of the vertex v is given by Here proj W ( x) is the orthogonal projection onto W and [ x] β is the coordinate vector of x under the basis β.
Example 2.4.1 Let L be the Laplacian matrix of the cubical graph given by The spectrum of its Laplacian is given by 0, 2, 2, 2, 4, 4, 4, 6. Let be the orthonormal basis of the 2-eigenspace. Then we have a three dimensional spectral drawing as follows.

Three Dimensional Real Representations
When X is the underlying graph of the skeleton of a prism, a Platonic solid, or an Archimedean solid X , one can show by cases that the second smallest Laplacian eigenvalues λ 2 is always of multiplicity three. (In other words, we have λ 2 = λ 3 = λ 4 < λ 5 .) In this case, the subspace W spanned by u 2 , u 3 , u 4 is the λ 2 -eigenspace. Let G be a group of symmetries of a given solid X , then G has a natural action on R[V ] by g( e v ) = e g(v) .
Proof: For all g ∈ G and v ∈ V , Therefore, two linear transformations g and L on R[V ] commute which implies that every eigenspace of L is g-invariant. Especially, the λ 2 -eigenspace W is g-invariant.
Corollary 2.5.2 The action of G restricted on W is a three dimensional real representation of G.
By Theorem 2.3.1, when X = Cay(G, S) is a rotational planar graph and X is not a circular graph, X is the underlying graph of the skeleton of some Platonic or Archimedean solid. Therefore, we can identify G as a group of isometries of S 2 by this manner.
Note that by Theorem 2.2.1 (δ i = 2(1 − cos θ i )) and first two assumptions, we always have θ 1 ≥ θ 2 . Since G is not a cyclic group, we also have u 1 = ± u 2 and sin ψ = 0.
Let α = { u 1 , u 2 , u 1 × u 2 }, which forms a basis of R 3 . For x ∈ R 3 , we will denote the coordinate vector (x 1 , x 2 , x 3 ) of x under the basis α by [ x] α . Note that under the basis α, the defining equation of S 2 becomes

Recall that
Let M 1 = 1 cos ψ cos ψ cos 2 ψ and M 2 = cos 2 ψ cos ψ cos ψ 1 , then we can rewrite the difference of the above two equations to obtain Therefore, we have it defines the union of two lines when δ 1 = δ 2 . In the formal case, the set D is the intersection of the unit sphere and a hyperbolic cylinder, which is the disjoint union of two so-called spherical ellipses. In the latter case, the set D is the intersection of the unit sphere and the union of two planes, which is the union of two great circles. We summarize the above result as the following theorem.

Maximal and minimal equal displacements
Now we would like to know how does the function vary on D. Since the above function only depends on x 1 , x 2 , it is sufficient to study the twovariables function f defined by on the following region: (Note that D 0 is the projection of D onto the plane spanned by u 1 and u 2 .) The extreme values of f (x) occur on either the critical points or the boundary points given by First, let us find the critical points of f (x). Set By the method of Lagrange multiplier, we have ∇f −λ∇F = 0, where λ is the Lagrange multiplier, which implies that The above equation has non-zero solution only when Recall that we assume that δ 1 ≥ δ 2 . If δ 1 = δ 2 , then there exists a unique solution (x 1 , x 2 ) = (0, 0). If δ 1 > δ 2 , then there exists no solution for b and we have Moreover, such a satisfies the condition a 2 sin 2 ψ ≤ 1 if and only if δ 1 cos 2 ψ ≤ δ 2 . In this case, the x 3 -coordinate of x is given by Let us summarize the above computation as the two following theorems.
Remark. The value δ 2 is the maximum of d s2 ( x), so it is the trivial upper bound of d s1 ( x) on D.
Next, let us find the values of f (x 1 , x 2 ) at the boundary points. Recall that the boundary points are given by By direct computation, Equation (4) becomes Subtracting δ 1 − δ 2 times of Equation (3) from Equation (4), we obtain which implies that This is a quite simple characterization of the boundary points. Plugging the above result into Equation (3), we obtain the following.
(Here we use the assumption that cos ψ ≤ 0.)

Isomorphism classes of X u
For two drawings X u and X v of X and A ∈ O(3), we say A is an isomorphism from X u and X v if there exists a permutation σ on G such that a) Ag u = σ(g) v, ∀ g ∈ G b) g 1 ∈ g 2 S if and only if σ(g 1 ) ∈ σ(g 2 )S, ∀ g 1 , g 2 ∈ G.
In this case, we say X u and X v are isomorphic, denoted by .
Example 3.4.1 Let A = −I and σ be the identity map, then for all g ∈ G, Ag u = −g u = σ(g)(− u). Thus −I ∈ Aut(X u ).
From the above examples, we see that Aut 0 (X) is a subgroup of Aut(X u ) for all u ∈ S 2 . Theorem 3.4.2 For u, v ∈ D, the following are equivalent.
Follow by the property that Aut 0 (X) is a subgroup of Aut(X u ).
a) ⇒ c): Since X v ∼ = X u , their edges are all of the same length, which implies that d s1 ( u) = d s2 ( v). By Proposition 3.4.1, there exists A ∈ Aut(X u ) so that A u = v. Since A preserves adjacency relation, we have {As u|s ∈ S} = {s v|s ∈ S}.
We have the following three cases.
1. When k 1 k 2 = 0, there are four solutions which are contained in one Aut 0 (X)-orbit.
3. When k 1 k 2 cos ψ = 0, there are eight solutions which are contained in two Aut 0 (X)-orbits.
It remains to show that when k 1 k 2 cos ψ = 0, then u and v must be in the same Aut 0 (X)-orbit. Suppose not, we may assume that which are two solutions of the above equations not contained in the same Aut 0 (X)-orbit.
Together with the assumption T u, T u = T u, T u , we should have T v, T v = T u, T u for all T of the form T + cI.
Then by direct computation, we have In all cases, T u, T u − T v, T v = 0, which is a contradiction.
It is easy to see that the four critical points in Theorem 3.3.2 lie in a single Aut 0 (X)-orbit and the four boundary points in Proposition 3.3.3 lie in two Aut 0 (X)-orbits. Together with Theorem 3.3.4, we conclude that: Corollary 3.4.3 For the displacement function d s1 ( x) on D, let δ max and δ min be the maximal value and the minimal value respectively. Then for the edge length δ max and δ min , there is a unique equilateral drawing X u up to isomorphism.

The angle between two edges
To identify the drawing X u with the skeleton of some uniform polyhedron, we should find the local configuration of X u . Let us compute the angle τ be between two edges u → s 1 u and u → s 2 u. Note that Besides, under the basis α = { u 1 , u 2 , u 1 × u 2 }, we have

Drawings of Schreier coset graphs
The drawing X u induces a drawing of the Schreier coset graph X H associated to (G/H, HSH/H) if u is fixed by all elements H. In this case, elements of H are rotations around u, which implies H is cyclic and it is generated by the rotation of the smallest angle. Therefore, we have the following proposition.
Therefore, u ∈ D if and only if s 1 ( u), u = s 2 ( u), u . Suppose s 1 ( u), u = s 2 ( u), u , then s 1 ( u) and s 2 ( u) lie in the same Stab( u)-orbit. Therefore, s 2 ( u) = g 1 s 1 ( u) for some g 1 ∈ Stab( u), which implies that g 2 := s 2 −1 g 1 s 1 fixes u. We conclude that s 2 = g 1 s 1 g 2 −1 ∈ Stab( u)s 1 Stab( u). Conversely, suppose s 2 = g 1 s 1 g 2 for some g 1 , g 2 ∈ Stab( u), then Since we can replace s 2 by s 2 −1 in the above theorem, we also have u ∈ D if and only if s 2 −1 ∈ Stab( u)s 1 Stab( u). On the other hand, when u is the rotational axis of h, h is a subgroup of Stab( u). Consequently, we obtain the following simple criterion. In general, to find all h in Corollary 3.6.3, one can only study by cases. On the other hand, some h exists for all cases.
Example 3.6.1 Let h 1 = s 1 s 2 , which is of order m 1 , then Therefore, a drawing of the Schreier coset graph X h1 can be induced by X u h 1 .
Example 3.6.2 Let h 2 = s 1 s 2 −1 which is of order m 2 , then Therefore, a drawing of the Schreier coset graph X h2 can be induced by X u h 2 .
By the same argument as the above two examples, for h 3 = s 1 −1 s 2 and h 4 = s 1 −1 s 2 −1 , we obtain drawings of Schreier coset graphs X h3 and X h4 respectively, namely X u h 3 and X u h 4 .
On the other hand, for A = R(π, u 3 / u 3 ) in Example 3.4.2, we have As 1 A −1 = s 1 −1 and As 2 A −1 = s 2 −1 , which implies that Therefore, X u h 1 and X u h 3 are isomorphic. Similarly, X u h 2 and X u h 4 are isomorphic.

Summarization
Let us summarize the results in this section sections. Set which is the edge length of X u . (Note that b < c since cos ψ < 0.) When X u is the unique shortest equilateral spherical drawing up to isomorphism. Its edge length is equal to √ b δ 1 δ 2 sin ψ and the angle τ between two edges u → s 1 u and u → s 2 u satisfies cos τ = cos θ 1 2 cos θ 2 2 + sin θ 1 2 sin θ 2 2 cos ψ.
We denote this vector by u min . For the longest drawing, we have the following.
In both cases, we denote this vector by u max . Besides, for h 1 = s 1 s 2 and h 2 = s 1 s 2 −1 , an equilateral drawing of the Schreier coset graph X hi can be induced by X u h i for i = 1, 2. Here u hi is a unit vector fixed by h i .

Rotational Planar Cayley Graphs
In this section, we study the rotational planar Cayley graphs Cay(G, S) introduced in Section 2.3, which are not circular. To do so, we will give a permutation representation of the group G so that one can use computer program to compute the real three dimensional representation σ of G introduced in Section 2.5 systematically. In fact, once we know cos ψ, we can set u 1 = (1, 0, 0) and u 2 = (cos ψ, sin ψ, 0).
For each Cayley graph, we list the value of cos ψ and four special equilateral spherical drawings X umin , X umax , X u h 1 and X u h 2 . Besides, for each drawing X u , the value of cos τ and the edge length are listed in the table. We can identify all except two drawings as the skeleton of some uniform polyhedron by computing the local configuration. To make the figures clearly, we add a sphere of suitable radius in the middle of the drawing and color the edges connecting to the point u as shown in the following figure.

The dihedral group D n
Let G = x, y where x = 1 n 2 (n − 1) · · · and y = (12 · · · n). Then G ∼ = D n . The following table is the result for n = 6.

Conclusion
From the results in Section 4.1 to Section 4.4, we see that the shortest equilateral drawing of a rotational planar Cayley graph gives us the skeleton of the corresponding uniform polyhedron. On the other hand, some longest equilateral drawings can not be identified as the skeleton of any uniform polyhedron and it is interesting to figure out these special structures.
As a final remark, we like to point out all vertex-transitive equilateral polyhedra can be realized by this manner if one consider all possible three representations of the underlying groups of rotational planar Cayley graphs.