### 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 |
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Pages (from-to) | 35-51 |

Number of pages | 17 |

Journal | Colloquium Mathematicum |

Volume | 132 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2013 Sep 30 |

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Hibino, Y., Lee, H. H., & Obata, N. (2013). Asymptotic spectral distributions of distance-k graphs of Cartesian product graphs.

*Colloquium Mathematicum*,*132*(1), 35-51. https://doi.org/10.4064/cm132-1-4