Graph Symmetry

The last decade has seen parallel developments in computer science and combinatorics, both dealing with networks having strong symmetry properties. Both developments are centred on Cayley graphs: in the design of large interconnection networks, Cayley graphs arise as one of the most frequently used models; on the mathematical side, they play a central role as the prototypes of vertex-transitive graphs. The surveys published here provide an account of these developments, with emphasis on the interplay of methods from group theory and graph theory that characterizes the subject. Topics covered include: combinatorial properties of various hierarchical families of Cayley graphs (fault tolerance, diameter, routing, forwarding indices, etc.) ; Laplace eigenvalues of graphs and their relations to forwarding problems, isoperimetric properties, partition problems, and random walks on graphs; vertex-transitive graphs of small orders and of orders having few prime factors; distance transitive graphs; isomorphism problems for Cayley graphs of cyclic groups; infinite vertex-transitive graphs (the random graph and generalizations, actions of the automorphisms on ray ends, relations to the growth rate of the graph).