M22 graph

The M₂₂ graph, also called the Mesner graph or Witt graph,^[1][2][3][4] is the unique strongly regular graph with parameters (77, 16, 0, 4).^[5] It is constructed from the Steiner system (3, 6, 22) by representing its 77 blocks as vertices and joining two vertices iff they have no terms in common or by deleting a vertex and its neighbors from the Higman–Sims graph.^[6][7]

M22 graph, Mesner graph[1][2][3]
Named after	Mathieu group M22, Dale M. Mesner
Vertices	77
Edges	616
Table of graphs and parameters

For any term, the family of blocks that contain that term forms an independent set in this graph, with 21 vertices. In a result analogous to the Erdős–Ko–Rado theorem (which can be formulated in terms of independent sets in Kneser graphs), these are the unique maximum independent sets in this graph.^[4]

It is one of seven known triangle-free strongly regular graphs.^[8] Its graph spectrum is (−6)²¹2⁵⁵16¹,^[6] and its automorphism group is the Mathieu group M22.^[5]

See also

• Cameron graph
• Higman–Sims graph
• Gewirtz graph