Upper bounding the coefficients of Tutte polynomials with a variable fixed

Abstract

The Tutte polynomial is a fundamental invariant of graphs and matroids. For a matroid M of rank r, the Tutte polynomial T_M(x,y) of M serves as a generalization of the characteristic polynomial chi_M(x), with the relationship given by chi_M(x) = (-1)^r T_M(1-x,0). For any real number t satisfying 0 <= t <= 1, under suitable exclusion hypotheses on M, we establish an upper bound on the coefficients of the polynomial T_M(x,t) (resp., T_M(t,y)) in terms of girth g and the number of circuits of size g (resp., maximum cardinality h of hyperplanes and the number of hyperplanes of size h). The resulting characteristic polynomial bound extends the bound of Delaplane and Wu [Discrete Math. 312(12-13) (2012) 2062-2068] to simple U_{g-1,g+1}-free matroids satisfying n >= r + c_g, thereby giving a partial affirmative answer to the question of extending their upper bound beyond binary matroids.

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BibTeX RIS

BibTeX

@misc{liu2026upper,
  title = {{Upper bounding the coefficients of Tutte polynomials with a variable fixed}},
  author = {Yuxin Liu and Weihua Yang and Xiaxia Guan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.34985},
  eprint = {2610.34985},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.34985},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Upper bounding the coefficients of Tutte polynomials with a variable fixed
AU  - Yuxin Liu
AU  - Weihua Yang
AU  - Xiaxia Guan
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.34985
UR  - https://arxiv.org/abs/2610.34985
AB  - The Tutte polynomial is a fundamental invariant of graphs and matroids. For a matroid M of rank r, the Tutte polynomial T_M(x,y) of M serves as a generalization of the characteristic polynomial chi_M(x), with the relationship given by chi_M(x) = (-1)^r T_M(1-x,0). For any real number t satisfying 0 <= t <= 1, under suitable exclusion hypotheses on M, we establish an upper bound on the coefficients of the polynomial T_M(x,t) (resp., T_M(t,y)) in terms of girth g and the number of circuits of size g (resp., maximum cardinality h of hyperplanes and the number of hyperplanes of size h). The resulting characteristic polynomial bound extends the bound of Delaplane and Wu [Discrete Math. 312(12-13) (2012) 2062-2068] to simple U_{g-1,g+1}-free matroids satisfying n >= r + c_g, thereby giving a partial affirmative answer to the question of extending their upper bound beyond binary matroids.
ER  -

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