Linear bound in terms of maxm axflow for the chromatic roots of series-parallel graphs

Gordon F. Royle; Alan D. Sokal

doi:10.1137/130930133

Linear bound in terms of maxm axflow for the chromatic roots of series-parallel graphs

Gordon F. Royle, Alan D. Sokal

Research output: Contribution to journal › Article

Abstract

We prove that the (real or complex) chromatic roots of a series-parallel graph with maxmaxflow ? lie in the disc |q - 1| < (∧ - 1)/log2. More generally, the same bound holds for the (real or complex) roots of the multivariate Tutte polynomial when the edge weights lie in the "real antiferromagnetic regime" - 1 ≤ v_e ≤ 0. For each ∧ ≥ 3, we exhibit a family of graphs, namely, the "leaf-joined trees", with maxmaxflow ? and chromatic roots accumulating densely on the circle |q - 1| = ∧ - 1, thereby showing that our result is within a factor 1/log 2 ≈ 1.442695 of being sharp.

Original language	English (US)
Pages (from-to)	2117-2159
Number of pages	43
Journal	SIAM Journal on Discrete Mathematics
Volume	29
Issue number	4
DOIs	https://doi.org/10.1137/130930133
State	Published - 2015

Keywords

Antiferromagnetic potts model
Chromatic polynomial
Chromatic roots
Maxmaxflow
Multivariate tutte polynomial
Series-parallel graph

ASJC Scopus subject areas

Mathematics(all)

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10.1137/130930133

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Royle, G. F., & Sokal, A. D. (2015). Linear bound in terms of maxm axflow for the chromatic roots of series-parallel graphs. SIAM Journal on Discrete Mathematics, 29(4), 2117-2159. https://doi.org/10.1137/130930133

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abstract = "We prove that the (real or complex) chromatic roots of a series-parallel graph with maxmaxflow ? lie in the disc |q - 1| < (∧ - 1)/log2. More generally, the same bound holds for the (real or complex) roots of the multivariate Tutte polynomial when the edge weights lie in the {"}real antiferromagnetic regime{"} - 1 ≤ ve ≤ 0. For each ∧ ≥ 3, we exhibit a family of graphs, namely, the {"}leaf-joined trees{"}, with maxmaxflow ? and chromatic roots accumulating densely on the circle |q - 1| = ∧ - 1, thereby showing that our result is within a factor 1/log 2 ≈ 1.442695 of being sharp.",

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AB - We prove that the (real or complex) chromatic roots of a series-parallel graph with maxmaxflow ? lie in the disc |q - 1| < (∧ - 1)/log2. More generally, the same bound holds for the (real or complex) roots of the multivariate Tutte polynomial when the edge weights lie in the "real antiferromagnetic regime" - 1 ≤ ve ≤ 0. For each ∧ ≥ 3, we exhibit a family of graphs, namely, the "leaf-joined trees", with maxmaxflow ? and chromatic roots accumulating densely on the circle |q - 1| = ∧ - 1, thereby showing that our result is within a factor 1/log 2 ≈ 1.442695 of being sharp.

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