On congruence in Zn and the dimension of a multidimensional circulant

✍ Scribed by M.A. Fiol

• Book ID: 103060915
• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 625 KB
• Volume: 141
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(94)00361-l

✦ Synopsis

From a generalization to Z" 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 ofn circulants with equal number of vertices p > 2, p a prime, has dimension n.