Partial Chebyshev Polynomials and Fan Graphs
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913769049817088 |
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| author | Młotkowski, Wojciech Obata, Nobuaki |
| author_facet | Młotkowski, Wojciech Obata, Nobuaki |
| contents | 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 $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_10697 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Partial Chebyshev Polynomials and Fan Graphs Młotkowski, Wojciech Obata, Nobuaki Combinatorics 05C50, 06C76, 51K99, 42C05 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 $n$. |
| title | Partial Chebyshev Polynomials and Fan Graphs |
| topic | Combinatorics 05C50, 06C76, 51K99, 42C05 |
| url | https://arxiv.org/abs/2412.10697 |