Asymptotic normality criteria of coefficients of a polynomial and their applications in combinatorics
葉永南教授中央研究院數學研究所 2:30 p.m. – 3:00 p.m. & 4:00 p.m. – 5:00 p.m. 鴻經館M107Abstract：
The asymptotic distribution
theory for coefficients of a polynomial is an active topic in asymptotic
analysis. In 1967, Harper proposed a criterion to measure the asymptotic
normality of a series of numbers, when he researched the asymptotic behavior of
Stirling numbers of the second kind. In this paper, we aim to develop some
further asymptotic normality criteria of coefficients of a polynomial with all
real roots or purely imaginary roots (including 0). These new asymptotic
normality criteria turn out to be very efficient and have abundant applications
in combinatorics, mainly including the coefficients of a series of
characteristic polynomials of adjacency matrix, Laplacian matrix, signless
Laplacian matrix, skew-adjacency matrix, chromatic polynomial, and some graph
numbers, such as matching numbers, independence numbers, clique numbers. Among
which, we generalize and verify some conjectures about asymptotic normality in
combinatorics, e.g., the matching numbers proposed by Godsil and Kahn , the
(signless) Laplacian coefficients claimed by Wang et al.