There has been further discussion of the G2 logo; here is a brief summary.
Elena Konstantinova tells me that it was designed by Vava Stefoglo; her portfolio is here. She is not a mathematician but received a good education in mathematics. She designed the logo to look like a faceted diamond.
Misha Klin told us about a survey paper he wrote with Christoph and Gerta Rücker and Gottfried Tinhofer on algebraic combinatorics in mathematical chemistry, published in Match (the journal of mathematical chemistry), volume 40 (1999), 7–138; you can find it here. The particular graph used in the logo (a circulant of order 8) is discussed, along with other circulants of order 8), in Example 8.16 on pages 99–100, but you might have to read quite a bit of the preceding material to see what is going on.
As I noted, it is an example of a Deza graph. It was discussed at previous G2 conferences, by Sergey Goryainov, Stefan Gyurki, Anton Betten and Yinfeng Zhu at G2R2, and by Sergei Lando at G2C2.