On Cayley graphs of {\bb Z}^4.

Igor A Baburin

doi:10.1107/s2053273320007159

Abstract

The generating sets of {\bb Z}^4 have been enumerated which consist of integral four-dimensional vectors with components -1, 0, 1 and allow Cayley graphs without edge intersections in a straight-edge embedding in a four-dimensional Euclidean space. Owing to computational restrictions the valency of enumerated graphs has been fixed to 10. Up to isomorphism 58 graphs have been found and characterized by coordination sequences, shortest cycles and automorphism groups. To compute automorphism groups, a novel strategy is introduced that is based on determining vertex stabilizers from the automorphism group of a sufficiently large finite ball cut out from an infinite graph. Six exceptional, rather `dense' graphs have been identified which are locally isomorphic to a five-dimensional cubic lattice within a ball of radius 10. They could be built by either interconnecting interpenetrated three- or four-dimensional cubic lattices and therefore necessarily contain Hopf links between quadrangular cycles. As a consequence, a local combinatorial isomorphism does not extend to a local isotopy.

Highlights

Cayley graphs provide a helpful tool to ‘visualize a group’ and to derive its properties in an essentially geometric way
We consider a Cayley graph of a group G with an inverse-closed finite generating set S as an undirected graph whose vertices correspond to group elements and vertices g; h 2 G are connected by an edge whenever gs 1⁄4 h; s 2 S
In this paper we provide a complete catalogue of Cayley graphs of Z4 with valency 10 which arise for generating sets of integral vectors with components À1, 0, 1

Summary

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. Loh, 2017). 3, 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). From Theorem 3 it is possible to determine the maximal valency for Cayley graphs of Zn which can be embedded in Rn without edge intersections provided the components of generating vectors are restricted to a certain range. [cf Power et al (2020), Proposition 4.5.] Let À be a Cayley graph of Zn with respect to a generating set S, and let À be embedded in Rn as described above with edges as straight-line segments. À is free of edge intersections (except at the vertices of À) iff hs; sÀ1 1i is a maximal rank 1 subgroup of Zn for any s1 2 S and hs; sÀ1 1; s2; sÀ2 1i is a maximal rank 2 subgroup of Zn for any s1 2 S and s2 2 S \ fs; sÀ1 1g

Computation of automorphism groups for Cayley graphs of Zn

Â C2 Â C2

Results and discussion