AI 中文总结
本文确定了交集图为圈时素数花束的部分Petrial多项式，并完整刻画了最低非零项次数为2的素数花束，进而给出完全二部图和完全三部图情形的推论。
AI 中文摘要
Gross、Mansour和Tucker [European J. Combin., 95 (2021): 103329] 引入了带状图 $G$ 的部分Petrial多项式，记为 $^{\partial}{\varepsilon^{\times}_G}(z)$。对于素数花束 $B_n$，Yan和Li [Discrete Appl. Math., 375 (2025): 281-289] 在交集图 $I(B_n)$ 为完全图或路径时确定了 $^{\partial}{\varepsilon^{\times}_{B_n}}(z)$，并给出了 $^{\partial}{\varepsilon^{\times}_{B_n}}(z)$ 中非零系数的最低次数为 $1$ 的等价条件。本文中，我们确定了交集图为圈时的 $^{\partial}{\varepsilon^{\times}_{B_n}}(z)$。此外，我们给出了其 $^{\partial}{\varepsilon^{\times}_{B_n}}(z)$ 中最低非零项次数为 $2$ 的素数花束的完整刻画，并确定了这些素数花束的部分Petrial多项式。作为推论，我们确定了当 $I(B_n)$ 为完全二部图和完全三部图时的 $^{\partial}{\varepsilon^{\times}_{B_n}}(z)$。
英文摘要
Gross, Mansour, and Tucker [European J. Combin., 95 (2021): 103329] introduced the \emph{partial Petrial polynomial} of a ribbon graph $G$, denoted by $^{\partial}{\varepsilon^{\times}_G}(z)$. For a prime bouquet $B_n$, Yan and Li [Discrete Appl. Math., 375 (2025): 281-289] determined $^{\partial}{\varepsilon^{\times}_{B_n}}(z)$ when the intersection graph $I(B_n)$ is either the complete graph or a path, and provided an equivalent condition under which the lowest degree of the nonzero coefficient in $^{\partial}{\varepsilon^{\times}_{B_n}}(z)$ is $1$. In this paper, we determine $^{\partial}{\varepsilon^{\times}_{B_n}}(z)$ when the intersection graph is a cycle. Moreover, we present a complete characterization of the prime bouquets whose lowest nonzero term in $^{\partial}{\varepsilon^{\times}_{B_n}}(z)$ is of degree $2$ and determine the partial Petrial polynomial for the prime bouquets. As corollaries, we determine $^{\partial}{\varepsilon^{\times}_{B_n}}(z)$ when $I(B_n)$ is the complete bipartite and tripartite graph.
