ABSTRACT
From its inception, the theory of groups has provided an interesting and powerful abstract approach to the study of the symmetries of various configurations. Any model of a given axiom system has an automorphism group, and graphs are no exception. It is observed that the group of a composite graph may be characterized in terms of the groups of its constituent graphs under suitable circumstances. Results are also presented on the existence of a graph with given group and given structural properties. There are several important operations on permutation groups which produce other permutation groups. This chapter explores four such binary operations: sum, product, composition, and power group. It discusses the group associated with a graph formed from other graphs by various operations. The group of a composite graph may often be expressed in terms of the groups of the constituent graphs.
