Graphs on topological spaces

In February this year I went to the AAA108 conference in Vienna. This was the 108th meeting of the Arbeitstagung Allgemeine Algebra. I was obviously going to talk about graphs on groups, but I thought I should take the time to see how the ideas would transfer to arbitrary algebras (in the sense of universal algebra). Some constructions obviously don’t; for example, the commuting graph is only available if the algebra has a distinguished binary operation. But a surprising amount of stuff works in complete generality. This includes graphs whose definition depends on the notion of the subalgebra generated by a subset; so the power graph, enhanced power graph, intersection power graph, generating graph, rank graph and independence graph can all be defined virtually without change. The only small problem occurs for the enhanced power graph. To define this graph on a group, we join x and y if and only if there is an element z such that both x and y lie in ⟨z⟩, or equivalently, if ⟨x,y⟩ is…