Connectivity of Cayley Graphs : A Special Family

Taking any finite group G, let H ⊂ G be such that 1 6∈ H (where 1 represents the identity element of G) and h ∈ H implies h−1 ∈ H. The Cayley graph X(G;H) is the graph whose vertices are labelled with the elements of G, in which there is an edge between two vertices g and gh if and only if h ∈ H. The exclusion of 1 from H eliminates the possibility of loops in the graph. The inclusion of the inverse of any element which is itself in H means that an edge is in the graph regardless of which endvertex is considered. It has been suggested that Cayley graphs should form good networks and several papers have been written about their fault tolerance. In particular, B. Alspach [2] proves certain results about the fault tolerance of a particular class of Cayley graphs, and isolates one family of graphs with interesting properties. The purpose of this paper is to exhibit some characteristics of graphs in this family. The fault tolerance of a graph is defined to be the largest number of vertices whose deletion cannot disconnect the graph. The connectivity of a graph is the smallest number of vertices whose deletion disconnects the graph. For a graph X, κ(X) denotes its connectivity. It is easy to see that the connectivity of a non-complete graph is always one more than its fault tolerance. The term connectivity will be used in this paper. An automorphism of a graph X is a permutation σ of the vertices of X with the property that if u and v are vertices of X, then there is an edge ∗This research was done while the author held an NSERC Undergraduate Research Award at Simon Fraser University.