graph is 5-ultrahomogeneous, it is ultrahomogeneous for every k; the only finite connected graphs of this type are complete graphs, Turán graphs, 3 × 3 rook's graphs, and the 5-cycle. The infinite Rado graph is countably ultrahomogeneous. There are only two connected graphs that are 4-ultrahomogeneous but not 5-ultrahomogeneous: the Schläfli graph and its complement. The proof relies on the classification of finite simple groups. See also Gosset graph - contains the Schläfli graph as an induced subgraph of the neighborhood of any vertex. Notes References . As cited by . . . As cited by . . . . . External links Andries E. Brouwer page. Individual graphs Regular graphs Strongly regular graphs