CYCLIC CONFIGURATIONS AND HAAR GRAPHS

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CYCLIC CONFIGURATIONS AND HAAR GRAPHS
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Haar graph of a natural number

• Let us write n in binary
• n bk-12k-1 bk-2 2k-2 ... b12 b0
• where B(n) (bk-1, bk-2, ..., b1, b0), bk-1
  1are binary digits of n. Graph H(n) H(k n)
  that is called the Haar graph of natural number
  n, has vertex set ui, vi, i0,1,...,k-1. Vertex
  ui is adjacent to vij, if and only if bj 1
  (arithmetic is mod k).

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Remark

• When defininig H(n) we assumed that k is the
  number of binary digits of n. In general for
  H(kn) one can take k to be greater than the
  number of binary digits. In such a case a
  different graph is obtained!

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Example

• Determine H(37).
• Binary digits
• B(37) 1,0,0,1,0,1
• k 6.
• H(37) H(637) is depicted on the left!

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Dipole qn

• Dipole qn has two vertices, joined by n parallel
  edges. If we want to distinguish the two
  vertices, we call one black, the other one white.
  On the left we see q5.
• Each dipole is a bipartite graph. Therefore each
  of its covering graphs is a bipartite graph.
• In particular q3 is a cubic graph also known as
  the theta graph q.

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Cyclic covers over a dipole

• Each Haar graph is a cyclic cover over a dipole.
  One can use the following recipe
• H(37) is determined by a natural number 37, or,
  equivalently by a binary sequence(1 0 0 1 0 1).
• The length is k6, therefore the group Z6.
• The indices are written below
• (1 0 0 1 0 1)
• (0 1 2 3 4 5)
• The 1s appear in positions 0, 3 in 5. These
  numbers are used as voltages for H(37).

0
3
5
Z6
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Exercises

• Graph on the left is the so-called Heawood graph
  H. Prove
• H is bipartite
• H is a Haar graph. (Find n!)
• Determine H as a cyclic cover over q3..
• Prove that H has no cycle of length lt 6.
• Prove that H is the smallest cubic graph of girth
  6.
• Find a hexagonal embedding of H in torus.
• Determine the dual of the embedded H.

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Heawood Graph in Torus

• On the left there is a hexagonal embedding of the
  Heawood graph in torus.

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Connected Haar graphs

• Graph G is connected if there is a path between
  aby two of its vertices.
• There exist disconnected Haar graphs, for
  instance H(10).
• Define n to be connected, if the corresponding
  Haar graph H(n) is connected.
• Disconnected numbers 2,4,8,10,16,32,34,36,40,42,6
  4...

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Exercises

• Prove that each 2m is a disconnected number.
• Show that the Möbius-Kantor graph G(8,3) is a
  Haar graph of some number. Which number is that?
• () Determine all generalized Petersen graphs
  that are Haar graphs of some natural number.
• Show that some Haar graphs are circulants.
• Show that some Haar graphs are non-circulants.

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Exercises, Continuation

• Prove that each Haar graph is vertex transitive.
• Prove that each Haar graph is a Cayley graph for
  a dihedral group.
• Prove that there exist bipartite Cayley graphs of
  dihedral groups that are not Haar graphs (such as
  the graph on the left).

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Exercises, The End

• The numbers n and m are cyclically equivalent,
  if the binary string of the first number can be
  cyclically transformed to the binary string of
  the second number. This means that the string
  can be cyclically permuted, mirrored or
  multiplied by a number relatively prime with the
  string length.
• The numbers n and m are Haar equivalent, if their
  Haar graphs are isomorphic H(n) H(m).
• Prove that cyclic equivalence implies Haar
  equivalence.
• Determine all numbers that are cyclically
  equivalent to 69.
• Use computer to show that 137331 and 143559 are
  Haar ekquivalent, but are not cyclically
  equivalent.

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The Mark Watkins Graph

• Cubic Haar graph H(536870930) has an interesting
  property. 536870930 is the smallest connected
  number that is cyclically equivalent to no odd
  number.
• Show that each Haar graph of an odd number
  H(2n1) is hamiltonian and therefore connected.

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Girth of Connected Haar graphs

• K2 is the only connected 1-valent Haar graph.
• Even cycles C2n are connected 2-valent Haar
  graphs.
• Theorem Let H be a connected Haar graph of
  valence d gt 2. Then either girth(H) 4 or
  girth(H) 6.

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Cyclic Configurations

• Symmetric (vr) configuration determined by its
  first column s of the configuration table where
  each additional column is obtained from s by
  addition (mod m) is called a cyclic
  configuration Cyc(ms).
• The left figure depicts a cyclic Fano
  configuration Cyc(71,2,4) Cyc(70,1,3).

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Connection to Haar graphs

• Theorem Symmetric configuration (vr), r 1 is
  cyclic, if and only if its Levi graph is a Haar
  graph with girth ¹ 4.
• Corollary Each cyclic configuration is point-
  and line-transitive and combinatorially
  self-dual.
• Corollary Each cyclic configuration (vr), r gt 2
  contains a triangle.
• Question Does there exist a cyclic configuration
  that is not combinatorially self-polar?

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Problem

• Study cyclic configurations with respect to flag
  orbits.