The cyclic index of adjacency tensor of generalized power hypergraphs

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