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1
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Asymptotic spectral distributions of distance k-graphs of large-dimensional hypercubes

100%
Obata N.
Banach Center Publications
|
2011
|
tom 96
|
nr 1
287-292
EN
We derive the asymptotic spectral distribution of the distance k-graph of N-dimensional hypercube as N → ∞.
2
Content available remote

Markov product of positive definite kernels and applications to Q-matrices of graph products

100%
Obata N.
Colloquium Mathematicum
|
2011
|
tom 122
|
nr 2
177-184
EN
We show positivity of the Q-matrix of four kinds of graph products: direct product (Cartesian product), star product, comb product, and free product. During the discussion we give an alternative simple proof of the Markov product theorem on positive definite kernels.
3
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Positive Q-matrices of graphs

100%
Obata N.
Studia Mathematica
|
2007
|
tom 179
|
nr 1
81-97
EN
The Q-matrix of a connected graph 𝒢 = (V,E) is $Q = (q^{∂(x,y)})_{x,y∈ V}$, where ∂(x,y) is the graph distance. Let q(𝒢) be the range of q ∈ (-1,1) for which the Q-matrix is strictly positive. We obtain a sufficient condition for the equality q(𝒢̃) = q(𝒢) where 𝒢̃ is an extension of a finite graph 𝒢 by joining a square. Some concrete examples are discussed.
4
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Asymptotic spectral analysis of growing graphs: odd graphs and spidernets

64%
Igarashi D., Obata N.
Banach Center Publications
|
2006
|
tom 73
|
nr 1
245-265
EN
Two new examples are given for illustrating the method of quantum decomposition in the asymptotic spectral analysis for a growing family of graphs. The odd graphs form a growing family of distance-regular graphs and the two-sided Rayleigh distribution appears in the limit of vacuum spectral distribution of the adjacency matrix. For a spidernet as well as for a growing family of spidernets the vacuum distribution of the adjacency matrix is the free Meixner law. These distributions are calculated through the Jacobi parameters obtained from structural data of graphs.
5
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Asymptotic spectral analysis of generalized Erdős-Rényi random graphs

51%
Liang S., Obata N., Takahashi S.
Banach Center Publications
|
2007
|
tom 78
|
nr 1
211-229
EN
Motivated by the Watts-Strogatz model for a complex network, we introduce a generalization of the Erdős-Rényi random graph. We derive a combinatorial formula for the moment sequence of its spectral distribution in the sparse limit.
6
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Asymptotic spectral distributions of distance-k graphs of Cartesian product graphs

51%
Hibino Y., Lee H., Obata N.
Colloquium Mathematicum
|
2013
|
tom 132
|
nr 1
35-51
EN
Let G be a finite connected graph on two or more vertices, and $G^{[N,k]}$ the distance-k graph of the N-fold Cartesian power of G. For a fixed k ≥ 1, we obtain explicitly the large N limit of the spectral distribution (the eigenvalue distribution of the adjacency matrix) of $G^{[N,k]}$. The limit distribution is described in terms of the Hermite polynomials. The proof is based on asymptotic combinatorics along with quantum probability theory.
7
Content available remote

Singleton independence

51%
Accardi L., Hashimoto Y., Obata N.
Banach Center Publications
|
1998
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tom 43
|
nr 1
9-24
EN
Motivated by the central limit problem for algebraic probability spaces arising from the Haagerup states on the free group with countably infinite generators, we introduce a new notion of statistical independence in terms of inequalities rather than of usual algebraic identities. In the case of the Haagerup states the role of the Gaussian law is played by the Ullman distribution. The limit process is realized explicitly on the finite temperature Boltzmannian Fock space. Furthermore, a functional central limit theorem associated with the Haagerup states is proved and the limit white noise is investigated.
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