Point-line collinearity graphs of two sporadic minimal parabolic geometries

The disc structure of the point-line collinearity graph for the rank two minimal parabolic geometries of the Thompson and Harada–Norton simple groups are investigated. Additionally details of the sub-orbits of these two groups in their conjugation action upon an involution conjugacy class is given. In particular, we give a list of representatives for each of these sub-orbits.