There has actually been mathematical research done on the structure of the Erdos graph. Erdos was certainly the most prolific mathematician of the last century, focusing primarily on combinatorics, but dabbling in just about everything. The number of papers which he was given co-author status is staggering.
The Erdős number, honouring the late Hungarian mathematician Paul Erdős, one of the most prolific writers of mathematical papers, is a way of describing the "collaborative distance", in regard to mathematical papers, between an author and Erdős.
Direct collaborators (worked with Erdős on the same paper) are assigned a number of 1. Someone who collaborated directly with a researcher with Erdős number 1, but not directly with Erdős, gets a number of 2. And so forth, with Erdős being the Kevin Bacon of the mathverse.
A small number of people are connected to both Erdős and Bacon and thus have a Erdős-Bacon number. One example is the actress Danica McKellar, best known for playing Winnie Cooper on the TV series, The Wonder Years. Her Erdős number is 4 and her Bacon number is 2.