The Fischer graph is one of examples; see page 569 of Suzuki, Michio. Group theory. II. Translated from the Japanese. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 248. Springer-Verlag, New York, 1986.

An special important case of the Fischer graph is the induced subgraph of the commuting graph of a group on a certain conjugacy class of involution so-called 3-traspositions.

The title of Section 10 of Chapter 6 of the above book of Suzuki is ``Graphs and Simple groups".