GRAPHS ON GROUPS

Abstract

The prototype for the graphs defined here is the commuting graph of a group, whose vertices are the group elements, two vertices being joined if they commute. A number of related graphs have been defined recently, including the power graph, generating graph, and independence graph. My purpose is to survey these graphs together, and indicate how they can be used to define old and new classes of groups, to characterise the groups by properties of the graphs, and in some cases to find beautiful and interesting graphs defined by groups.