harmonic functions and the chromatic polynomial
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Nov 16, 2022 •177 likes •456 views

harmonic functions and the chromatic polynomial R. Kenyon (Brown) based on joint work with A. Abrams, W. Lam The chromatic polynomial G ( n ) of a graph G is the number of proper colorings with n colors. (adjacent vertices have di ff erent

  1. harmonic functions and the chromatic polynomial R. Kenyon (Brown) based on joint work with A. Abrams, W. Lam

  2. The chromatic polynomial χ G ( n ) of a graph G is the number of proper colorings with n colors. (adjacent vertices have di ff erent colors) χ ( n ) = n ( n − 1)( n − 2) χ satisfies a contraction-deletion rule: χ G ( n ) = χ G − e ( n ) − χ G/e ( n ) but is # P -hard to compute in general.

  3. The Dirichlet problem 1 A graph G = ( V, E ) c : E → R > 0 the edge conductances 4 2 B ⊂ V boundary vertices 3 u : B → R boundary values Find f : V → R harmonic on V \ B 6 5 and f | B = u . X 0 = ∆ f ( x ) = c e ( f ( x ) − f ( y )) 0 y ∼ x f is the unique function with f | B = u minimizing the Dirichlet energy X c e ( f ( x ) − f ( y )) 2 E ( f ) = e = xy { edge energy

  4. A harmonic function induces a compatible orientation: an acyclic orientation with no internal sources or sinks, and no oriented paths from lower boundary values to higher boundary values. “current flows downhill” 3 1 2 4 6 9 5 7 8 10 12 11 We let Σ be the set of compatible orientations How many are there?

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Then ¯ [ F = F σ σ ∈ Σ where F σ = { f ∈ F | sign( d f ) = σ } . The F σ are convex polytopes. y 1 1.0 0.8 0.6 x y 0.4 0.2 x 0 0.2 0.4 0.6 0.8 1.0

  6. Fixed energy problem:

  7. Can we adjust edge conductances so that all bulbs burn with the same brightness?

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