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

• Main Authors: Morales, John Vincent S., Obata, Nobuaki, Tanaka, Hajime
• Format: text
• Published: Animo Repository 2020
• Subjects: Graph theory; Hypergeometric series; Orthogonal polynomials; Hilbert space; Central limit theorem; Mathematics
• Online Access: https://animorepository.dlsu.edu.ph/faculty_research/2567
• Institution: De La Salle University

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.