The Zara graph is the unique graph on 126 vertices satisfying the properties that 1) every maximal clique (of which there are a total of 567) has six vertices, and 2) that if is a maximal clique and is a vertex outside of , then has exactly two neighbors in (Blokhuis and Brouwer 1984).
The graph is strongly regular with parameters , though it is not the unique graph having such parameters. It is also both distance-regular with intersection array and distance-transitive.
It has graph spectrum and is therefore an integral graph. It has graph automorphism group order .
It is a Hamiltonian graph.
The Zara graph is implemented in the Wolfram Language as GraphData["ZaraGraph"].