图联接下$\tau$-多项式的实根性
Real-rootedness of the $τ$-polynomial under graph joins
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中文总结 AI 辅助
本文证明若两个不相交简单图的τ-多项式仅有实零点，则其联图的τ-多项式也仅有实零点，解决了Brenti等人1994年提出的猜想。
中文摘要 AI 辅助
对于具有$n$个顶点的简单图$G$，将其色多项式在上升阶乘基中写为$$ \chi_G(x)=\sum_{i=0}^{n}(-1)^{n-i}c_i(G)\langle x\rangle_i,$$其中$ \langle x\rangle_i=x(x+1)\cdots(x+i-1).$ 相关的$\tau$-多项式$$ \tau_G(x)=\sum_{i=0}^{n}c_i(G)x^i $$由Brenti于1992年定义并系统研究。本文证明：若两个顶点不相交的简单图$G$和$H$的$\tau$-多项式仅有实零点，则它们的联图$G\vee H$的$\tau$-多项式也仅有实零点。这解决了Brenti、Royle和Wagner自1994年以来提出的猜想。
英文摘要
For a simple graph $G$ with $n$ vertices, write its chromatic polynomial in the rising factorial basis as $$ χ_G(x)=\sum_{i=0}^{n}(-1)^{n-i}c_i(G)\langle x\rangle_i,$$ where $ \langle x\rangle_i=x(x+1)\cdots(x+i-1).$ The associated $τ$-polynomial $$ τ_G(x)=\sum_{i=0}^{n}c_i(G)x^i $$ was defined and systematically investigated by Brenti in 1992. In this paper, we prove that if the $τ$-polynomials of two vertex-disjoint simple graphs $G$ and $H$ have only real zeros, then the $τ$-polynomial of their join $G\vee H$ has only real zeros. This settles a conjecture posed by Brenti, Royle and Wagner since 1994.
发表机构
- Center for Combinatorics, LPMC, Nankai University（南开大学组合数学中心，LPMC）
- School of Mathematics, Tianjin University（天津大学数学学院）
- Department of Mathematics, Tianjin University of Finance and Economics（天津财经大学数学系）
- College of Mathematical Sciences & Institute of Mathematics and Interdisciplinary Sciences, Tianjin Normal University（天津师范大学数学科学学院及数学与交叉科学研究院）
机构由 AI 辅助整理，请以论文原文为准。
