[FOM] Reverse mathematics of spectral graph theory
Timothy Y. Chow
tchow at alum.mit.edu
Tue Sep 6 17:37:16 EDT 2011
Some months ago, someone asked on cstheory.SE for examples of statements
in graph theory that can be proved only by spectral means.
http://cstheory.stackexchange.com/questions/5099/proofs-obtained-only-through-spectral-graph-theory
It's not so easy to come up with definitive examples, because it is
sometimes possible to unpack the linear algebra into a combinatorial proof
that makes no explicit reference to linear-algebraic concepts. For
example, Sundar Vishwanathan posted a preprint to the ArXiv not long ago
giving a combinatorial proof of the Graham-Pollak theorem, which for a
long time was an example of a purely graph-theoretic statement that seemed
to require linear algebra (if not eigenvalues explicitly):
http://arxiv.org/abs/1007.1553
Vishwanathan makes some intriguing comments at the end that, roughly
speaking, suggest that fast-growing functions might be needed to prove the
Graham-Pollak theorem. "Fast-growing" here doesn't mean Ackerman and
beyond but just exponential versus polynomial.
This makes me wonder whether there is, at least potentially, some way to
prove that certain classical theorems of spectral graph theory (like the
Graham-Pollak theorem, the friendship theorem, the Hoffman-Singleton
theorem, etc.) "require" some strong assumption in a reverse-mathematical
sense. I know that Cook and others have looked in the proof complexity of
linear algebra and have isolated some candidates for hard tautologies
coming from linear algebra. Perhaps some of these spectral graph theorems
are "equivalent" in some sense one of these hard tautologies? Has anyone
looked into this kind of thing?
Tim
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