AI 中文总结
该研究将路径和圈的图多面体的不等式替换为分段形式，利用自伴算子及其特征值得到相关集合体积的收敛级数，还得出路径与对应圈关联集合体积相差常数因子α的结论。
AI 中文摘要
我们研究路径和圈的图多面体的一种变体，其中我们将不等式$x_{i} + x_{i+1} \leq 1$替换为两个不等式：当$0 \leq x_{i} \leq \alpha$时，$(1-\alpha) \cdot x_{i} + \alpha \cdot x_{i+1} \leq \alpha$；当$\alpha \leq x_{i} \leq 1$时，$\alpha \cdot x_{i} + (1-\alpha) \cdot x_{i+1} \leq \alpha$。利用自伴算子及其特征值，我们得到了这些集合体积的收敛级数。作为推论，我们得到：与$n$个顶点的路径相关联的集合的体积，和与$n+1$个顶点的圈相关联的集合的体积，相差一个常数因子$\alpha$。
英文摘要
We study a variant of the graph polytopes of a path and of a cycle where we replace the inequality $x_{i} + x_{i+1} \leq 1$ with the two inequalities $(1-α) \cdot x_{i} + α\cdot x_{i+1} \leq α$ for $0 \leq x_{i} \leq α$ and $α\cdot x_{i} + (1-α) \cdot x_{i+1} \leq α$ for $α\leq x_{i} \leq 1$. Using a self-adjoint operator and its eigenvalues we obtain convergent series for their volumes. As a corollary we obtain that the volumes of the set associated to a path on $n$ vertices and the set associated to a cycle on $n+1$ vertices are related by a constant factor of $α$.
Comments9 pages, 1 figure
Journal refThe Australasian Journal of Combinatorics 96(1) (2026), 38--48
