On Cayley graphs of

Cayley graphs of with valency 10 have been enumerated which correspond to generating sets of integral vectors with components −1, 0, 1 and which are embedded in a four-dimensional Euclidean space without edge intersections.


Introduction
Cayley graphs provide a helpful tool to 'visualize a group' and to derive its properties (e.g. defining relations) in an essentially geometric way (cf. Lö h, 2017). As usual, we consider a Cayley graph of a group G with an inverse-closed finite generating set S (such that S = S À1 6 3 1) as an undirected graph whose vertices correspond to group elements and vertices g; h 2 G are connected by an edge whenever gs ¼ h; s 2 S. For additive groups (e.g. for Z n , n being a positive integer) we may write s ¼ h À g.
Cayley graphs of crystallographic groups in a Euclidean plane were treated in detail in a well known book by Coxeter & Moser (1980). The situation in higher dimensions (> 2) is far from having been completely explored. Although in dimension 3 different enumeration methods indeed produced many Cayley graphs of crystallographic groups relevant to structural chemistry (e.g. Fischer, 1974Fischer, , 1993, their potential has never been used in full [some applications are described by Eon (2012)]. Despite some results on lattice nets or bouquet nets (Delgado-Friedrichs & O'Keeffe, 2009;Moreira de Oliveira & Eon, 2014), the terms adopted by crystallographers for Cayley graphs of Z n , complete enumerations for Z 3 (under fairly natural assumptions) have become available only recently (Power et al., 2020). Many important properties of Cayley graphs of Z n were derived by Kostousov (2007) which we quote below (Section 2.1).
There exists only one (up to isomorphism) Cayley graph of Z 4 with valency 8 that corresponds to a four-dimensional hypercubic lattice. In this paper we provide a complete catalogue of Cayley graphs of Z 4 with valency 10 which arise for generating sets of integral vectors with components À1, 0, 1 (loosely speaking, the shortest) and which are embedded in a four-dimensional Euclidean space, i.e. free of edge crossings in a straight-edge embedding (in other words, edges intersect at most at common vertices). The restriction to valency 10 in the present study is due to a significant increase in the computational demand for isomorphism checking already for the next possible valency 12, in which case effective invariants are to be developed to quickly distinguish non-isomorphic graphs. The structure of graphs is characterized in terms of coordination sequences and shortest cycles. Additionally, we apply a novel strategy to compute automorphism groups.
2. Theoretical background and computational methodology 2.1. Some properties of Cayley graphs of Z n We start by summarizing important facts about Cayley graphs of Z n . In the following, we associate Z n with an additive group of n-dimensional integral vectors of an affine Euclidean space R n .
Theorem 1. [Kostousov (2007), Theorem 3, part (a), and Proposition 3.] Let S and M be generating sets of Z n which consist of n-dimensional integral row vectors. The respective Cayley graphs are isomorphic iff there is a matrix X 2 GLðn; ZÞ with |det(X)| = 1 such that M ¼ SX.
Theorem 1 provides a handy criterion for isomorphism testing by solving a system of linear equations. We note that isomorphism testing by computing canonical forms according to Delgado-Friedrichs (2004) turns out to be rather expensive in dimensions n > 3.
As an immediate consequence we obtain that vertex stabilizers in Aut(À) are finite.
Any Cayley graph of Z n allows a natural embedding in R n , with vertices as nodes of an integral lattice and edges as straight-line segments corresponding to generators. Any automorphism of a graph in this embedding is induced by an affine map of R n . The following theorem provides a grouptheoretic condition for when this embedding is free of edge intersections.
Theorem 3. [cf. Power et al. (2020), Proposition 4.5.] Let À be a Cayley graph of Z n with respect to a generating set S, and let À be embedded in R n as described above with edges as straight-line segments. Then À is free of edge intersections (except at the vertices of À) iff hs 1 ; s À1 1 i is a maximal rank 1 subgroup of Z n for any s 1 2 S and hs 1 ; s À1 1 ; s 2 ; s À1 2 i is a maximal rank 2 subgroup of Z n for any s 1 2 S and s 2 2 S \ fs 1 ; s À1 1 g.

Proof.
A pair of intersecting edges of À spans a one-or twodimensional affine subspace. give rise to square grids which are shifted against each other in a two-dimensional plane defined by s 1 ; s 2 that forces edges to cross. As a consequence, edge intersections do not take place iff subgroups H (1) and H (2) are maximal subgroups of Z n with rank 1 and 2, respectively. & Remark 1. If the conditions of Theorem 3 are fulfilled but subsets of S generate non-maximal subgroups of Z n with rank d ! 3, then an affine subspace of dimension d accommodates a finite number of connected components (each of dimensionality d) which do not cross each other. This implies the existence of Hopf links between the cycles of a graph (cf. Section 3).
Remark 2. From Theorem 3 it is possible to determine the maximal valency for Cayley graphs of Z n which can be embedded in R n without edge intersections provided the components of generating vectors are restricted to a certain range. For example, if vectors with components À1, 0, 1 are considered, the maximal valency is 6, 14, 30, 62, 126 for n = 2, 3, 4, 5, 6, respectively.

Computation of automorphism groups for Cayley graphs of Z n
Since any Cayley graph À of Z n obviously does not show up vertex collisions in a barycentric placement, the method of Delgado-Friedrichs (2004) (2003) for a less formal exposition] can be used to compute Aut(À). Here we have adopted a different strategy that involves a computation of a vertex stabilizer in Aut(À) from the local structure of a graph À. This strategy is quite general and appears to be very effective for vertex-transitive periodic graphs with finite vertex stabilizers in Aut(À).
To facilitate the following discussion, let us establish some notation for graphs and group actions on various sets associated with them.
Let À be a connected simple graph with finite valencies of vertices. The distance between vertices x and y, d(x, y), is defined as the number of edges in a shortest path from x to y.
The automorphism group of À, Aut(À), is regarded as a group of all adjacency-preserving permutations on the vertex set of À. Two (generally non-isomorphic) vertex-transitive graphs À 1 and À 2 are said to be locally isomorphic within a ball of radius r if hB À 1 ðx; rÞi ffi hB À 2 ðy; rÞi. If G is a permutation group on a set X, then Stab G (x) = {g | xg = x, g 2 G}, i.e. the stabilizer of an element Proposition 1. [Trofimov (2012), Section 3.] Let À be a connected vertex-transitive graph and v be a vertex of À. For any non-negative integer r there exists a minimal integer (r) ! r such that Stab AutðhBðv; ðrÞÞiÞ ðvÞ Bðv; rÞ ¼ Stab AutðÀÞ ðvÞ Bðv; rÞ : In other words, any automorphism of hB(v, r)i fixing v which can be extended to an automorphism of hB(v, (r))i can also be extended to an automorphism of À.
Proposition 2. Let À be a Cayley graph of Z n . For any vertex v of À, the stabilizer Stab Aut(À) (v) acts faithfully on B(v, 1).
Proof. By Theorem 2 Aut(À) is isomorphic to a crystallographic group. Vertices adjacent to v form an n-dimensional convex hull that cannot be stabilized pointwise by any crystallographic isometry (or an affine map) of R n . & Proposition 3. Let À be a Cayley graph of a group G with respect to a generating set S, and v be a vertex of À. Then Aut(À) = hS, Stab Aut(À) (v)i.
For a Cayley graph À of Z n Propositions 1 and 2 allow us to determine a faithful action of Stab Aut(À) (v) on B(v, 1) as a permutation group from the restriction of Stab Aut(hB(v, (1))i) (v) to B(v, 1). A practical computation of (1), Stab Aut(À) (v) and eventually Aut(À) (the latter requires Proposition 3) is facilitated by employing the fact that any automorphism of À is induced by an affine map of R n if vertices of À are associated with nodes of an integral lattice as done in Section 2.1.
(i) For some k (1) generate a finite subgraph hB(v, k)i of À and compute Aut(hB(v, k) (iii) Check (by solving systems of linear equations) if permutations computed at step (ii) are induced by integral n Â n matrices. If so, then k = (1) is found. The set T of the so-obtained matrices generates an integral matrix representation of Stab Aut(À) (v), and we proceed to step (iv). Otherwise we set k: = k + 1 and go back to step (i).
(iv) Aut(À) is output as a matrix group generated by S and T: Aut(À) = hS, Ti (elements of S and T are expressed here as (n+1) Â (n+1) augmented matrices).
Remark. Recently another method [see Section 3.1 in Bremner et al. (2014)] has come to our attention that allows one to compute Stab Aut(À) (v) [v = (0, . . . , 0)] as an integral matrix group by making use of a positive definite symmetric matrix Q ¼ P i s i s T i , where the sum runs over column vectors s i 2 S; i ¼ 1; . . . jSj. The automorphism group of Q, Aut(Q), is defined as AutðQÞ ¼ fX 2 GLðn; ZÞ j XQX T ¼ Qg and corresponds to the isometry group of an n-dimensional integral lattice with a Gram matrix Q. Aut(Q) can be computed using the algorithm of Plesken & Souvignier (1997) as implemented in the AUTO program. 2 Stab Aut(À) (v) is readily obtained as a setwise stabilizer of S in Aut(Q).

Results and discussion
With the above theory in mind, we have implemented in the GAP programming language (GAP, 2019) the search for generating sets of Z 4 which give rise to Cayley graphs of valency 10 by enumerating quintuples of four-dimensional vectors with components À1, 0, 1. Filtering out generating sets which satisfy our Theorem 3 (Section 2.1) and yield isomorphic graphs was done on the fly, and the computation of automorphism groups was implemented in a separate program making use of nauty (McKay, 2009) and the Cryst package (Eick et al., 2019). For checking purposes, automorphism groups were also computed with an alternative method based on the Remark in Section 2.2. The results are gathered in Table 1. Furthermore, the supporting information contains explicit lists of generators, point symbols (Blatov et al., 2010) and coordination sequences up to the 15th sphere.
To our knowledge, only three out of the 58 graphs have been known before, namely, #1, #2 and #20 which correspond to primitive hexagonal tetragonal, I-centred cubic orthogonal and primitive icosahedral lattices 3 (O' Keeffe, 1995). The 'topological' diversity of the graphs is very much restricted since they all turn out to be closely related to a fivedimensional (primitive) cubic lattice as shown by point symbols and coordination sequences. This is not accidental since Cayley graphs of Z n with valency 2(n+1) are indeed quotients of an (n+1)-dimensional cubic lattice with respect to some rank 1 subgroup (cf. Eon, 2011). Low-dimensional quotients necessarily inherit certain properties from their parent higher-dimensional counterparts.
Generating sets for 55 graphs (all except #1, #2 and #20) contain subsets corresponding to non-maximal Z 3 or Z 4 subgroups (or both). This means that quadrangular cycles of the graphs are linked (cf. Remark 1 to Theorem 3). Let us discuss this phenomenon in more detail for six exceptional graphs (#49, #52, #54, #55, #56, #58, see Table 2) which are locally isomorphic to a five-dimensional cubic lattice within a ball of radius 10, as proven by isomorphism computations. Subgraph enumeration for #49, #52, #54, #55, #56, #58 has identified sets of three-as well as four-dimensional cubic lattices (last column of Table 2). These sets contain a finite number of connected components which interpenetrate each other in a manner as shown in Fig. 1  (3); (3, 5, 7, 9) † The notation (a, b, c, . . . ) means that different subsets are possible which contain a (or b, or c, . . . ) connected components.
1 In actual computations, if a graph in question is a quotient of some other graph (as is the case for 10-valent Cayley graphs of Z 4 which are quotients of a five-dimensional hypercubic lattice, cf. Section 3), a good initial guess for k is the radius starting from which coordination sequences of a graph and its quotient become different. Then the described procedure for finding (r) converges rather rapidly. For the graphs from Table 1 computations yield (1) 13.
interpenetrating cubic lattices in a fourth dimension. Alternatively, they could be viewed as interconnected interpenetrating four-dimensional cubic lattices. Obviously both constructions imply Hopf links between quadrangular cycles. Qualitatively speaking, Hopf links arise from keeping the same amount (40) of quadrangular cycles per vertex while reducing the number of coordinate two-dimensional planes from 10 (in five dimensions) to 6 (in four dimensions). It is clear that quadrangular cycles of a five-dimensional cubic lattice lie separately in orthogonal two-dimensional coordinate planes and therefore are not linked. As a consequence, although being locally isomorphic to a five-dimensional cubic lattice, the above graphs are not locally isotopic to it. These examples illustrate perhaps a general phenomenon that knotting in crystal structures can formally arise from projections of high-dimensional periodic nets and represents a compromise of how a high-dimensional object could fit into a lower-dimensional space.