The generating graph of a finite group

Abstract

The generating graph of a finite group \(G\) is a graph whose vertices are the elements of \(G\), and with an edge between \(x\) and \(y\) if and only if \(x\) and \(y\) generate \(G\). This is clearly only an interesting object for groups that are \(2\)-generated, but fortunately a great many interesting families of groups are \(2\)-generated, including all finite simple groups. I’ll give a survey of what is currently known about the generating graph, and finish with several open problems.