On congruence in Z^n and the dimension of a multidimensional circulant

TL;DR

This paper explores the structure of multidimensional circulants, relating their graph dimensions to the Smith normal form of matrices, and fully characterizes certain circulants and their Cartesian products.

Contribution

It establishes a connection between the Smith normal form of matrices and the dimensions of multidimensional circulants, providing a complete characterization of 2-step circulants and their Cartesian products.

Findings

Dimension of Cartesian product of n circulants with prime order p is n.

Fully characterized 2-step multidimensional circulants that are circulants.

Linked graph dimension to the Smith normal form of associated matrices.

Abstract

From a generalization to $Z^n$ of the concept of congruence we define a family of regular digraphs or graphs called multidimensional circulants, which turn out to be Cayley (di)graphs of Abelian groups. This paper is mainly devoted to show the relationship between the Smith normal form for integral matrices and the dimensions of such (di)graphs, that is the minimum ranks of the groups they can arise from. In particular, those 2-step multidimensional circulants which are circulants, that is Cayley (di)graphs of cyclic groups, are fully characterized. In addition, a reasoning due to Lawrence is used to prove that the cartesian product of $n$ circulants with equal number of vertices $p>2$, $p$ a prime, has dimension $n$.