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 the adjacency tensor of G. Let m , s be positive integers such that s ≥ 2 and m = s t. 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 ⋅ c (G). In this paper, by using a matrix equation over 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. [ABSTRACT FROM AUTHOR]