Partial Chebyshev Polynomials and Fan Graphs

2024/12/14 by Wojciech Młotkowski, Młotkowski, Wojciech, Nobuaki Obata +1 · 1 citation
Mathematics · #05C50 #06C76 #42C05 #51K99 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.2412.10697

openalex publication_date 2024/12/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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^e_n(x) and U^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 K₁+P_n, we derive a new polynomial ϕ_n(x) which is factorized by partial Chebyshev polynomial U^e_n(x). We prove that QEC(K₁+P_n) is given in terms of the minimal zero of ϕ_n(x), and obtain the explicit value of QEC(K₁+P_n) for an even n and its reasonable exstimate for an odd n.

Cited by

• Quadratic Embedding Constants of Corona Graphs

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