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.