Partial Chebyshev Polynomials and Fan Graphs

TL;DR

This paper introduces partial Chebyshev polynomials $U^{\mathrm{e}}_n(x)$ and $U^{\mathrm{o}}_n(x)$, establishing their factorization with the classical Chebyshev polynomials and deriving key identities. It then leverages these polynomials to compute the quadratic embedding constant $ ext{QEC}(K_1+P_n)$ for fan graphs, by constructing a polynomial $\phi_n(x)$ that factors as $(x-1)U^{\mathrm{e}}_n(x)S_n(x)$ and whose minimal zero $\alpha_n$ determines $ ext{QEC}(K_1+P_n)$ via $\text{QEC}(K_1+P_n)=-2\alpha_n-2$. The main results give explicit values for even $n$ and tight estimates for odd $n$, and show that $ ext{QEC}(K_1+P_n)$ forms a strictly increasing sequence converging to $0$, thereby advancing the classification of graphs by the QE embedding criterion. The approach highlights a deep connection between Chebyshev-type polynomials and distance-geometry of graphs, enabling simplified evaluation of QEC for a fundamental graph family.

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 $φ_n(x)$ 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 $φ_n(x)$, and obtain the explicit value of $\mathrm{QEC}(K_1+P_n)$ for an even $n$ and its reasonable exstimate for an odd $n$.

Figures (1)

• Figure 1: Fan graph $K_1+P_n$ with $n=5$

Theorems & Definitions (45)

• Theorem 2.1
• Theorem 2.2
• Remark 2.3
• Proposition 2.4
• Remark 2.5
• Theorem 2.6
• proof
• Theorem 2.7
• proof
• Theorem 2.8