The Average Connectivity of Graphs of Automorphisms of

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The Average Connectivity of Graphs of
Automorphisms of

• Daniel Franz

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Average Connectivity

• Let G(V,E) be an undirected graph.
• Define ?G(u,v) to be the number of disjoint paths
  between u and v.
• The average connectivity of G is

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Graphs Based on Inverses mod p

• G(V,E).
• V0,1,,p-1, p prime.
• Every element x in V is connected to x1, x-1,
  and x-1, all operations mod p.
• A loop is attached at 0.

p23
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Automorphisms of z

• All automorphisms of have the form
  fk(x)xk (mod p).
• , with fk(x)
  k.

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Graphs Based on Automorphisms

• Gp,k(V,E), V0,1,...,p-1, p prime.
• Connect every element x in V to x1, x-1, and
  fk(x).
• Counting loops and multiple edges, Gp,k is a
  4-regular graph.

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Examples
G71,17
G71,43
G41,3
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Involutions and Noninvolutions

• An automorphism fk is an involution if it is of
  order 2 i.e. if k21 (mod p-1).

An involution with p31, k19
A noninvolution with p31, k17
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Equivalent Graphs

• If k1k21 (mod p-1), then Gp,k1 is isomorphic to
  Gp,k2.

G47,11
G47,21
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Loop and Involution Indices

• Gp,k has gcd(p-1,k-1)1 loops and
  gcd(p-1, k2-1)1 vertices involved in
  involution pairs or loops.
• Loop index
  .
• Involution index
  .
• If p2 (mod 3) and p3 (mod 4), then
  
  I(Gp,k)2/(p-1).

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Examples
L(G47,11)I(G47,11) 1/23
L(G37,11)1/18 I(G37,11)1/3
L(G37,13)I(G37,13)1/3
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Paths in Gp,k

• A chord is an interior edge a chord component is
  a collection of intersecting chords.

Chord component
Chord component
Chord component
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Paths in Gp,k

• A chord path between vertices u and v is a
  sequence of chords that can be used to travel
  from u to v.

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Paths in Gp,k

• Let u, v be vertices of G. If there are two
  disjoint chord paths leading from u to v, then
  there are 4 disjoint paths from u to v.

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Component Size

• Fix k and consider only primes p for which
  k is in .
• Then for any component C with vertices only in
  the top half of Gp,k, CO(p1/2).

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Average Connectivity of Gp,k

• Fix k and consider only primes p for which
  k is in .

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Acknowledgments

• Patrick Bahls
• Sam Kaplan
• Dave Peifer