Partial Chebyshev Polynomials and Fan Graphs

TL;DR

This paper introduces partial Chebyshev polynomials, explores their properties, and applies them to compute the quadratic embedding constant of fan graphs, providing explicit formulas and estimates.

Contribution

It presents new partial Chebyshev polynomials, derives their properties, and applies them to calculate the quadratic embedding constant of fan graphs.

Findings

Derived basic properties and relations of partial Chebyshev polynomials.

Established a formula for the quadratic embedding constant of fan graphs.

Provided explicit values and estimates for the quadratic embedding constant.

Abstract

Motivated by the product formula of the Chebyshev polynomials of the second kind $U_n(x)$, we newly introduce the partial Chebyshev polynomials $U^{\mathrm{e}}_n(x)$ and $U^{\mathrm{o}}_n(x)$ and derive their basic properties, relations to the classical Chebyshev polynomials, and new factorization formulas for $U_n(x)$. In order to calculate the quadratic embedding constant (QEC) of a fan graph $K_1+P_n$, we derive a new polynomial $\phi_n(x)$ which is factorized by partial Chebyshev polynomial $U^{\mathrm{e}}_n(x)$. We prove that $\mathrm{QEC}(K_1+P_n)$ is given in terms of the minimal zero of $\phi_n(x)$, and obtain the explicit value of $\mathrm{QEC}(K_1+P_n)$ for an even $n$ and its reasonable exstimate for an odd $n$.