TY - JOUR

T1 - Asymptotic joint spectra of Cartesian powers of strongly regular graphs and bivariate Charlier–Hermite polynomials

AU - Morales, John Vincent S.

AU - Obata, Nobuaki

AU - Tanaka, Hajime

N1 - Funding Information:
NO was supported by JSPS KAKENHI Grant Number JP16H03939, and HT was supported by JSPS KAKENHI Grant Numbers JP25400034 and JP17K05156.
Publisher Copyright:
© Instytut Matematyczny PAN, 2020.

PY - 2020

Y1 - 2020

N2 - Generalizing previous work of Hora (1998) on the asymptotic spectral analysis for the Hamming graph H(n, q) which is the nth Cartesian power Kq□n of the complete graph Kq on q vertices, we describe the possible limits of the joint spectral distribution of the pair (G□n, G□n) of the nth Cartesian powers of a strongly regular graph G and its complement G, where we let n → ∞, and G may vary with n. This result is an analogue of the bivariate central limit theorem, and we obtain in this way the bivariate Poisson distributions and the standard bivariate Gaussian distribution, together with the product measures of univariate Poisson and Gaussian distributions. We also report a family of bivariate hypergeometric orthogonal polynomials with respect to the last distributions, which we call the bivariate Charlier–Hermite polynomials, and prove basic formulas for them. This family of orthogonal polynomials seems previously unnoticed, possibly because of its peculiarity.

AB - Generalizing previous work of Hora (1998) on the asymptotic spectral analysis for the Hamming graph H(n, q) which is the nth Cartesian power Kq□n of the complete graph Kq on q vertices, we describe the possible limits of the joint spectral distribution of the pair (G□n, G□n) of the nth Cartesian powers of a strongly regular graph G and its complement G, where we let n → ∞, and G may vary with n. This result is an analogue of the bivariate central limit theorem, and we obtain in this way the bivariate Poisson distributions and the standard bivariate Gaussian distribution, together with the product measures of univariate Poisson and Gaussian distributions. We also report a family of bivariate hypergeometric orthogonal polynomials with respect to the last distributions, which we call the bivariate Charlier–Hermite polynomials, and prove basic formulas for them. This family of orthogonal polynomials seems previously unnoticed, possibly because of its peculiarity.

KW - Central limit theorem

KW - Hypergeometric series

KW - Orthogonal polynomial

KW - Spectral distribution

KW - Strongly regular graph

KW - quantum probability

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U2 - 10.4064/cm7724-7-2019

DO - 10.4064/cm7724-7-2019

M3 - Article

AN - SCOPUS:85086925367

VL - 162

SP - 1

EP - 22

JO - Colloquium Mathematicum

JF - Colloquium Mathematicum

SN - 0010-1354

IS - 1

ER -