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Let T = T (x 0) be the Terwilliger algebra of the Johnson scheme J (N , D) , where x 0 is a fixed base point. In Liang et al. (2017), an observation is made on the parameters of Leonard systems that arise from the irreducible representations of T. This paper accounts for the observation from the viewpoint of group representations. In particular, it is shown that T is isomorphic to the centralizer algebra for the stabilizer of the base point x 0 in the automorphism group of J (N , D). [ABSTRACT FROM AUTHOR]

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]

Abstract: Let be a nonsingular connected mixed graph. We determine the mixed graphs on at least seven vertices with exactly two Laplacian eigenvalues greater than 2. In addition, all mixed graphs with exactly one Laplacian eigenvalue greater than 2 are also characterized. [Copyright &y& Elsevier]

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]