The stabilizing index and cyclic index of coalescence and Cartesian product of uniform hypergraphs

Yi-Zheng Fan; Meng-Yu Tian; Min Li

paper

The stabilizing index and cyclic index of coalescence and Cartesian product of uniform hypergraphs

arXiv:2002.11715

Abstract

Let $G$ be connected uniform hypergraph and let $A (G)$ be the adjacency tensor of $G$ . The stabilizing index of $G$ is the number of eigenvectors of $A (G)$ associated with the spectral radius, and the cyclic index of $G$ is the number of eigenvalues of $A (G)$ with modulus equal to the spectral radius. Let $G_{1} ⊙ G_{2}$ and $G_{1} □ G_{2}$ be the coalescence and Cartesian product of connected $m$ -uniform hypergraphs $G_{1}$ and $G_{2}$ respectively. In this paper, we give explicit formulas for the the stabilizing indices and cyclic indices of $G_{1} ⊙ G_{2}$ and $G_{1} □ G_{2}$ in terms of those of $G_{1}$ and $G_{2}$ or the invariant divisors of their incidence matrices over $Z_{m}$ , respectively.