Asymptotic spectral distributions of distance-k graphs of Cartesian product graphs

Yuji Hibino; Hun Hee Lee; Nobuaki Obata

doi:10.4064/cm132-1-4

Asymptotic spectral distributions of distance-k graphs of Cartesian product graphs

Yuji Hibino, Hun Hee Lee, Nobuaki Obata

Center for Data-driven Science and Artificial Intelligence (CDS)

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

Abstract

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.

Original language	English
Pages (from-to)	35-51
Number of pages	17
Journal	Colloquium Mathematicum
Volume	132
Issue number	1
DOIs	https://doi.org/10.4064/cm132-1-4
Publication status	Published - 2013

Keywords

Adjacency matrix
Cartesian product graph
Central limit theorem
Distance-k graph
Hermite polynomials
Quantum probability
Spectrum

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10.4064/cm132-1-4

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AU - Lee, Hun Hee

AU - Obata, Nobuaki

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N2 - 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.

AB - 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.

KW - Adjacency matrix

KW - Cartesian product graph

KW - Central limit theorem

KW - Distance-k graph

KW - Hermite polynomials

KW - Quantum probability

KW - Spectrum

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Asymptotic spectral distributions of distance-k graphs of Cartesian product graphs

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Keywords

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