Prime graph

In the mathematics of graph theory and finite groups, a prime graph is an undirected graph defined from a group. These graphs were introduced in a 1981 paper by J. S. Williams, credited to unpublished work from 1975 by K. W. Gruenberg and O. Kegel.

Definition
The prime graph of a group has a vertex for each prime number that divides the order (number of elements) of the given group, and an edge connecting each pair of prime numbers $$p$$ and $$q$$ for which there exists a group element with order $$pq$$.

Equivalently, there is an edge from $$p$$ to $$q$$ whenever the given group contains commuting elements of order $$p$$ and of order $$q$$, or whenever the given group contains a cyclic group of order $$pq$$ as one of its subgroups.

Properties
Certain finite simple groups can be recognized by the degrees of the vertices in their prime graphs. The connected components of a prime graph have diameter at most five, and at most three for solvable groups. When a prime graph is a tree, it has at most eight vertices, and at most four for solvable groups.

Related graphs
Variations of prime graphs that replace the existence of a cyclic subgroup of order $$pq$$, in the definition for adjacency in a prime graph, by the existence of a subgroup of another type, have also been studied. Similar results have also been obtained from a related family of graphs, obtained from a finite group through the degrees of its characters rather than through the orders of its elements.