Geodesic
Asymptotic spectral distributions of distance-$k$ graphs of Cartesian product graphs
Yuji Hibino1 ; Hun Hee Lee2 ; Nobuaki Obata3
1 Department of Mathematics Saga University Saga, 840-8502, Japan
2 Department of Mathematical Sciences and Research Institute of Mathematics Seoul National University Seoul 151-747, Republic of Korea
3 Graduate School of Information Sciences Tohoku University Sendai, 980-8579, Japan
Colloquium Mathematicum, Tome 132 (2013) no. 1, pp. 35-51.

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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 $k\ge 1$, 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.
DOI : 10.4064/cm132-1-4
Keywords: finite connected graph vertices distance k graph n fold cartesian power fixed obtain explicitly large limit spectral distribution eigenvalue distribution adjacency matrix limit distribution described terms hermite polynomials proof based asymptotic combinatorics along quantum probability theory

Yuji Hibino 1 ; Hun Hee Lee 2 ; Nobuaki Obata 3

1 Department of Mathematics Saga University Saga, 840-8502, Japan
2 Department of Mathematical Sciences and Research Institute of Mathematics Seoul National University Seoul 151-747, Republic of Korea
3 Graduate School of Information Sciences Tohoku University Sendai, 980-8579, Japan 
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Yuji Hibino; Hun Hee Lee; Nobuaki Obata. Asymptotic spectral distributions
 of distance-$k$ graphs of Cartesian product graphs. Colloquium Mathematicum, Tome 132 (2013) no. 1, pp. 35-51. doi : 10.4064/cm132-1-4. http://geodesic.mathdoc.fr/articles/10.4064/cm132-1-4/

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