# The cyclic index of adjacency tensor of generalized power hypergraphs

## TL;DR

This paper investigates the cyclic index of adjacency tensors in generalized power hypergraphs, disproves a conjecture about its behavior, and provides conditions and characterizations related to the index.

## Contribution

The paper disproves the conjecture that the cyclic index scales linearly in generalized power hypergraphs and offers new conditions and characterizations for when the conjecture holds.

## Key findings

- Counterexample to the conjecture that c(G^{m,s})=s * c(G)
- Sufficient conditions identified for the conjecture to hold
- An equivalent matrix equation characterization over _m provided

## Abstract

Let $G$ be a $t$-uniform hypergraph, and let $c(G)$ denote the cyclic index of the adjacency tensor of $G$. Let $m,s,t$ be positive integers such that $t \ge 2$, $s \ge 2$ 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 $c(G^{m,s})=s \cdot c(G)$. In this paper we show that the conjecture is false by giving a counterexample, and give some sufficient conditions for the conjecture holding. Finally we give an equivalent characterization of the equality in the conjecture by using a matrix equation over $\mathbb{Z}_m$.