Let $G$ be a $t$-uniform hypergraph, and let $c\left(G\right)$ denote the cyclic index of the adjacency tensor of $G$. Let $m,s$ be positive integers such that $s\ge 2$ and $m=st$. The generalized power $G^{m,s}$ of $G$ is obtained from $G$ by blowing up each vertex into an $s$-set and preserving the adjacency relation. It was conjectured that $c\left(G^{m,s}\right)=s\cdot c\left(G\right)$. In this paper, by using a matrix equation over ${\mathbb{Z}}_m$ that characterizes the spectral symmetry or cyclic index of an $m$-uniform hypergraph, we give an equivalent condition for the equality in the conjecture. Finally we show that the conjecture is false by a counterexample.

中文翻译：

---

广义幂超图的邻接张量的循环索引

让 $G$ 成为 $Ť$-统一的超图，并让 $C（G）$ 表示的邻接张量的循环索引 $G$。让$米，s$ 是正整数，这样 $s\ge 2$ 和 $米=sŤ$。广义幂$G^{米，s}$ 的 $G$ 从获得 $G$ 通过将每个顶点炸成一个 $s$设置并保留邻接关系。据推测$C（G^{米，s}）=s\cdot C（G）$。在本文中，通过在${\mathbb{ž}}_{米}$ 表征光谱的对称性或循环指数 $米$-一致超图，我们为猜想中的相等给出一个等价条件。最后，通过反例证明该猜想是错误的。