Partial Chebyshev Polynomials and Fan Graphs

Abstract

Motivated by the product formula of the Chebyshev polynomials of the second kind $U_n(x)$, we newly introduce the partial Chebyshev polynomials $U_n^{\mathrm{e}}(x)$ and $U_n^{\mathrm{o}}(x)$ 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 $\ddot{I}{•}_n(x)$ which is factorized by partial Chebyshev polynomial $U_n^{\mathrm{e}}(x)$. We prove that $\mathrm{QEC}(K_1+P_n)$ is given in terms of the minimal zero of $\ddot{I}{•}_n(x)$, and obtain the explicit value of $\mathrm{QEC}(K_1+P_n)$ for an even $n$ and its reasonable exstimate for an odd $n$.