Automorphism groups of rational circulant graphs through the use of Schur rings

TL;DR

This paper investigates the automorphism groups of rational circulant graphs, using Schur rings to relate the problem to orthogonal group block structures and expressing the groups via generalized wreath products.

Contribution

It introduces a novel approach linking automorphism groups of rational circulant graphs to orthogonal group block structures through Schur rings.

Findings

Automorphism groups characterized as generalized wreath products of symmetric groups

Equivalence established between automorphism groups and orthogonal group block structures

Method provides a systematic way to determine automorphism groups of rational circulant graphs

Abstract

The paper concerns the automorphism groups of Cayley graphs over cyclic groups which have a rational spectrum (rational circulant graphs for short). With the aid of the techniques of Schur rings it is shown that the problem is equivalent to consider the automorphism groups of orthogonal group block structures of cyclic groups. Using this observation, the required groups are expressed in terms of generalized wreath products of symmetric groups.