Real roots of the flow polynomial
For every graph , let be the chromatic polynomial of and let be the flow polynomial of . If is loopless, then for all sufficiently large integers (as = # of k-colorings of ). It follows from Seymour's 6-flow theorem that if has no bridge, then for all integers (as = # of nowhere-zero flows in the group of integers modulo ). It is natural to ask if all real roots of these polynomials are small. For the chromatic polynomial, , this is not the case. There exist graphs with chromatic number 3 for which has arbitrarily large real roots. The above conjecture asserts that the flow polynomial exhibits the opposite behavior. One word of caution, it is known that the set of roots of flow polynomials is dense in the complex plane.
[S] P.D. Seymour, Nowhere-Zero 6-Flows, J. Combinatorial Theory Ser. B 30 (1981) 130-135. MathSciNet
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