Propagation in Graphs Modulo a Prime

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Propagation in Graphs Modulo a Prime

• Jacob Steinhardt

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Background

• Given a graph X, adjacency matrix A(X), its
  spectrum is defined to be all of the eigenvalues
  of the adjacency matrix, that is all k for which
  Avkv has a solution for non-zero vectors v.
• The spectrum of a graph over the complex numbers
  gives us bounds regarding many of the properties
  of a graph.

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My Idea

• What information can the spectrum of a graph over
  finite fields yield?
• One answer it can tell us about the behavior of
  the graph modulo p a prime (and, perhaps for
  composite n as well).
• Theorem If A is a linear operator on a
  finite-dimensional vector space V (over an
  arbitrary field F), then A is nilpotent iff all
  of its eigenvalues are zero.

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One Case A Grid

• Consider a graph consisting of all points
  (x1,...,xk ) with 0xidi, where two points are
  connected if all but one of the coordinates are
  the same, and the remaining coordinate differs by
  1. We call this a d1d2dk grid.
• Theorem The d1dk grid is nilpotent mod p if
  and only if one of the following hold
• p2,and each di is either one less than a power
  of 2 or 2. Furthermore, the number of indices i
  with di2 is even.
• di1 for all i.

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Nilpotency

• Note nilpotency mod p is equivalent to saying
  that, for any starting assignment of numbers to
  each vertex, if we repeatedly replace the number
  at each vertex with the sum of the numbers on the
  neighboring vertices, all numbers will eventually
  be divisible by p.
• Other possible questions When can we say that
  the number at a single vertex will always be
  eventually divisible by p?
• Can we tie the spectrum mod p to some property
  that exists outside of mod p?

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Other Questions

• Eigenvalue techniques over the complex numbers
  order the eigenvalues by magnitude. Can we define
  some sort of norm on the eigenvalues over finite
  fields?
• Idea Let dF(x,y) F(x)F(y)F-F(x)nF(y)F.
  Is this a meaningful distance? Would have to
  prove that F(x)F(y)F-F(x)nF(y)FF(x-y)F
  or possibly F(x-y)F-1.
• Possibly x F(x)F could define a sort of
  distance.