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Let G be connected uniform hypergraph and let A (G) be the adjacency tensor of G. The stabilizing index of G is exactly the number of eigenvectors of A (G) associated with the spectral radius, and the cyclic index of G is exactly 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 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. [ABSTRACT FROM AUTHOR]

Let Γ (x 0) be a finite rooted tree, for which Γ is the underlying tree and x 0 the root. Let T be the Terwilliger algebra of Γ with respect to x 0. We study the structure of the principal T -module. As a result, it is shown that T recognizes the isomorphism class of Γ (x 0). [ABSTRACT FROM AUTHOR]