Coefficients of univariate Tutte polynomials with one variable fixed
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| Format: | Preprint |
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2025
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| _version_ | 1866916041725050880 |
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| author | Ma, Tianlong Guan, Xiaxia Jin, Xian'an |
| author_facet | Ma, Tianlong Guan, Xiaxia Jin, Xian'an |
| contents | It is well known that the 2-variable Tutte polynomials contain chromatic polynomial and flow polynomial of graphs, i.e. the cases of $y=0$ and $x=0$. In 2013, Kálmán introduced the interior and exterior polynomials which generalized the cases of $y=1$ and $x=1$ of Tutte polynomials of graphs to hypergraphs, and further polymatroids. There have been some results on coefficients of these polynomials, which motivate us to study uniformly the coefficients of $T_M(x,t)$ and $T_M(t,y)$, where $M$ is a matroid and $t$ is a fixed real number. In this paper, we introduce two mutually dual parameters $f_k(M)$ and $g_k(M)$ ($g_1(M)$ is the grith of $M$) for any nonnegative integer $k$, obtaining: (1) Formulas for coefficients of the higher-degree terms (related to $g_2(M)$ and $f_2(M)$, respectively) of $T_M(x,t)$ and $T_M(t,y)$ in terms of circuits and hyperplanes of $M$; (2) when $0\leq t \leq 1$, coefficients of the more higher-degree terms (related to $g_1(M)$ and $f_1(M)$, respectively) of $T_M(x,t)$ and $T_M(t,y)$ are further simplified and characterized; (3) As applications, some known results in the cases $t=0$ and $t=1$ are derived and generalized, and the unimodality of these coefficients in (1) are proved when $t\leq 1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_06095 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Coefficients of univariate Tutte polynomials with one variable fixed Ma, Tianlong Guan, Xiaxia Jin, Xian'an Combinatorics 05C31, 05B35 It is well known that the 2-variable Tutte polynomials contain chromatic polynomial and flow polynomial of graphs, i.e. the cases of $y=0$ and $x=0$. In 2013, Kálmán introduced the interior and exterior polynomials which generalized the cases of $y=1$ and $x=1$ of Tutte polynomials of graphs to hypergraphs, and further polymatroids. There have been some results on coefficients of these polynomials, which motivate us to study uniformly the coefficients of $T_M(x,t)$ and $T_M(t,y)$, where $M$ is a matroid and $t$ is a fixed real number. In this paper, we introduce two mutually dual parameters $f_k(M)$ and $g_k(M)$ ($g_1(M)$ is the grith of $M$) for any nonnegative integer $k$, obtaining: (1) Formulas for coefficients of the higher-degree terms (related to $g_2(M)$ and $f_2(M)$, respectively) of $T_M(x,t)$ and $T_M(t,y)$ in terms of circuits and hyperplanes of $M$; (2) when $0\leq t \leq 1$, coefficients of the more higher-degree terms (related to $g_1(M)$ and $f_1(M)$, respectively) of $T_M(x,t)$ and $T_M(t,y)$ are further simplified and characterized; (3) As applications, some known results in the cases $t=0$ and $t=1$ are derived and generalized, and the unimodality of these coefficients in (1) are proved when $t\leq 1$. |
| title | Coefficients of univariate Tutte polynomials with one variable fixed |
| topic | Combinatorics 05C31, 05B35 |
| url | https://arxiv.org/abs/2503.06095 |