### Abstract

The generalized theta graph χ_{S1},⋯,S_{k} consists of a pair of endvertices joined by k internally disjoint paths of lengths S_{1,}⋯,Sk≥1. We prove that the roots of the chromatic polynomial π(χ_{S1},⋯,S_{k}, Z) of a k-ary generalized theta graph all lie in the disc z - 1 ≤ [ 1 + o (1) ]k/log k, uniformly in the path lengths S_{i}. Moreover, we prove that χ_{2},⋯,2≃K_{2 }indeed has a chromatic root of modulus [1+0(1)]k/log k. Finally, for k ≤ 8 we prove that the generalized theta graph with a chromatic root that maximizes z - 1 is the one with all path lengths equal to 2; we conjecture that this holds for all k.

Original language | English (US) |
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Pages (from-to) | 272-297 |

Number of pages | 26 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 83 |

Issue number | 2 |

DOIs | |

State | Published - 2001 |

### Keywords

- Chromatic polynomial
- Chromatic roots
- Complete bipartite graph
- Generalized theta graph
- Graph
- Lambert W function
- Potts model
- Series-parallel graph

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

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## Cite this

Brown, J. I., Hickman, C., Sokal, A. D., & Wagner, D. G. (2001). On the chromatic roots of generalized theta graphs.

*Journal of Combinatorial Theory. Series B*,*83*(2), 272-297. https://doi.org/10.1006/jctb.2001.2057