AI 中文总结
本文针对循环循环图的约化谱函数，构造了首一零化多项式，确定了最小递推阶数的充分条件，对特定集合推导了极小九次零化多项式。
AI 中文摘要
设$S$为非空有限正整数集合，记$q = \max S$，当$n>2q$时，$B_n(S)$为循环循环图$G_n(S)$对应的切比雪夫型多项式的归一化乘积序列。当$G_n(S)$连通时，$B_n(S)$既是归一化生成树数，也是约化谱函数的归一化特殊值，该函数是由非平凡邻接谱构造的行列式型函数。已知固定步长循环生成树序列存在切比雪夫根表示与线性递推，本文从这些表示出发，显式构造了$\mathbb Z[X]$中首一零化多项式$\mathcal H_S(X)$，其次数为$3^{q-1}$，它给出了$B_n(S)$递推阶数的一般上界。通过收集重合的指数基并考虑可能的抵消，确定了最小零化多项式，给出了最小递推阶数恰好为$3^{q-1}$的充分条件；对$S=\{1,2,3\}$和$S=\{1,3\}$，显式推导了对应的九次零化多项式并证明其极小性。
英文摘要
Let $S$ be a nonempty finite set of positive integers, let $q=\max S$, and let $B_n(S)$, $n>2q$, be the normalized product sequence associated with the Chebyshev-type polynomial of the cyclic circulant graph $G_n(S)$. When $G_n(S)$ is connected, $B_n(S)$ is both the normalized spanning-tree number and a normalized special value of the \emph{reduced spectral function}, a determinant-type function constructed from the non-trivial adjacency spectrum. Chebyshev root representations and the existence of linear recurrences for fixed-step circulant spanning-tree sequences are known. Starting from these representations, we explicitly construct a monic annihilating polynomial $\mathcal H_S(X)\in\mathbb Z[X]$ of degree $3^{q-1}$, which yields a general upper bound for the recurrence order of $B_n(S)$. By collecting coincident exponential bases and accounting for possible cancellations, we determine the minimal annihilating polynomial and give a sufficient condition under which the minimal recurrence order is exactly $3^{q-1}$. For $S=\{1,2,3\}$ and $S=\{1,3\}$, we explicitly derive the corresponding ninth-degree annihilating polynomials and prove their minimality.
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