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Linear bounds on characteristic polynomials of matroids

Published online by Cambridge University Press:  10 January 2019

SUIJIE WANG ,
YEONG–NAN YEH  and
FENGWEI ZHOU
SUIJIE WANG
Affiliation:
Institute of Mathematics, Hunan University, China. e-mail: wangsuijie@hnu.edu.cn
YEONG–NAN YEH
Affiliation:
Institute of Mathematics, Academia Sinica, Taiwan. e-mail: mayeh@math.sinica.edu.tw
FENGWEI ZHOU
Affiliation:
Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong. e-mail: fzhou@connect.ust.hk
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Abstract

Let χ(t) = a0tn – a1tn−1 + ⋯ + (−1)rartn−r be the chromatic polynomial of a graph, the characteristic polynomial of a matroid, or the characteristic polynomial of an arrangement of hyperplanes. For any integer k = 0, 1, …, r and real number x ⩾ k − r − 1, we obtain a linear bound of the coefficient sequence, that is

\begin{align*} {r+x\choose k}\leqslant \sum_{i=0}^{k}a_{i}{x\choose k-i}\leqslant {m+x\choose k}, \end{align*}
where m is the size of the graph, matroid, or hyperplane arrangement. It extends Whitney’s sign-alternating theorem, Meredith’s upper bound theorem, and Dowling and Wilson’s lower bound theorem on the coefficient sequence. In the end, we also propose a problem on the combinatorial interpretation of the above inequality.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 168 , Issue 3 , May 2020 , pp. 505 - 518
Copyright
Copyright © Cambridge Philosophical Society 2019

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Footnotes

†

This work was supported by the National Natural Science Foundation of China (11871204).

References

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