This is the unique graph on 126 vertices with the properties that (i) any maximal clique has six vertices, and (ii) if $C$ is a maximal clique and $v$ is a vertex outside of $C$, then $v$ has exactly two neighbours in $C$. Uniqueness was shown by Blokhuis and Brouwer (1984). It is a strongly regular graph with parameters $(126,45,12,18)$, but there are other examples of SRGs with these parameters.

Number of vertices: | $126$ |

Diameter: | $2$ |

Intersection array: | $\{45,32;1,18\}$ |

Spectrum: | $45^1 3^{90} (-9)^{35}$ |

Automorphism group: | $PSU(4,3).2^2$ |

Distance-transitive: | $\text{Yes}$ |

Primitive |

- $\text{Adjacency matrix}$
- $\text{Adjacency matrix in GAP format}$
- $\text{Adjacency matrix in CSV format}$
- $\text{Graph in GRAPE format}$

- A. Blokhuis and A. E. Brouwer, Uniqueness of a Zara graph on 126 points and non-existence of a completely regular two-graph on 288 points, 1984.