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M22 graph, Mesner graph[1][2][3]
Named afterMathieu group M22, Dale M. Mesner
Vertices77
Edges616
Table of graphs and parameters

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]

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

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References

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  1. ^ a b "Mesner graph with parameters (77,16,0,4). The automorphism group is of order 887040 and is isomorphic to the stabilizer of a point in the automorphism group of NL2(10)"
  2. ^ a b Slide 5 list of triangle-free SRGs says "Mesner graph"
  3. ^ a b Section 3.2.6 Mesner graph
  4. ^ a b Godsil, Christopher; Meagher, Karen (2015), "Section 5.4: The Witt graph", Erdős–Ko–Rado Theorems: Algebraic Approaches, Cambridge Studies in Advanced Mathematics, Cambridge University Press, pp. 94–96, ISBN 9781107128446
  5. ^ a b Brouwer, Andries E. “M22 Graph.” Technische Universiteit Eindhoven, http://www.win.tue.nl/~aeb/graphs/M22.html. Accessed 29 May 2018.
  6. ^ a b Weisstein, Eric W. “M22 Graph.” MathWorld, http://mathworld.wolfram.com/M22Graph.html. Accessed 29 May 2018.
  7. ^ Vis, Timothy. “The Higman–Sims Graph.” University of Colorado Denver, http://math.ucdenver.edu/~wcherowi/courses/m6023/tim.pdf. Accessed 29 May 2018.
  8. ^ Weisstein, Eric W. “Strongly Regular Graph.” From Wolfram MathWorld, mathworld.wolfram.com/StronglyRegularGraph.html.
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📚 Artikel Terkait di Wikipedia

M22

M22, M.22 or M-22 may refer to: BFW M.22, prototype, 1928 German bomber Jancsó-Szokolay M22, 1937 Hungarian sailplane Magni M-22 Voyager, autogyro Mooney

Higman–Sims graph

group of automorphisms of the Hoffman–Singleton graph. Take the M22 graph, a strongly regular graph srg(77,16,0,4) and augment it with 22 new vertices

Mathieu group M22

group theory, the Mathieu group M22 is a sporadic simple group of order    443,520 = 27 · 32 · 5 · 7 · 11 ≈ 4×105. M22 is one of the 26 sporadic groups

Strongly regular graph

graph on GQ(2, 4). The Hoffman–Singleton graph is an srg(50, 7, 0, 1). The Gewirtz graph is an srg(56, 10, 0, 2). The M22 graph aka the Mesner graph is

Cameron graph

strongly regular graphs on which the Mathieu group M22 acts as symmetries taking every vertex to every other vertex. The smaller M22 graph is another. Brouwer

List of finite simple groups

time it was thought that the full covering group of M22 was 6⋅M22. In fact J4 has no subgroup 12⋅M22.) Z. Janko (1976). "A new finite simple group of order

Higman–Sims group

isomorphic to M22. HS is the simple subgroup of index two in the group of automorphisms of the Higman–Sims graph. The Higman–Sims graph has 100 nodes

Fischer group

whose symmetry group is M22. A basic set generates an abelian group of order 210, which extends in Fi22 to a subgroup 210:M22. The next Fischer group