Cayley graphs and symmetric interconnection networks

TL;DR

This paper explores automorphism groups of Cayley graphs and their role in designing fault-tolerant interconnection networks, analyzing specific graph families like hypercubes and Cayley graphs from linear codes.

Contribution

It provides new insights into automorphism groups of Cayley graphs and their application to optimizing fault-tolerance in interconnection networks.

Findings

Vertex-connectivity of edge-transitive graphs is maximized.

Automorphism groups of specific Cayley graph families are characterized.

Open problems in automorphism groups and network design are discussed.

Abstract

These lecture notes are on automorphism groups of Cayley graphs and their applications to optimal fault-tolerance of some interconnection networks. We first give an introduction to automorphisms of graphs and an introduction to Cayley graphs. We then discuss automorphism groups of Cayley graphs. We prove that the vertex-connectivity of edge-transitive graphs is maximum possible. We investigate the automorphism group and vertex-connectivity of some families of Cayley graphs that have been considered for interconnection networks; we focus on the hypercubes, folded hypercubes, Cayley graphs generated by transpositions, and Cayley graphs from linear codes. New questions and open problems are also discussed.