Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2412.10697

Partial Chebyshev Polynomials and Fan Graphs

Wojciech Młotkowski·Nobuaki Obata
View original source ↗Read source document ↗Export metadata ↓

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 $n$.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Wojciech Młotkowski, Nobuaki Obata. 2025-04-01. Partial Chebyshev Polynomials and Fan Graphs. https://arxiv.org/abs/2412.10697

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On nut graphs with two vertex and three edge orbits

Nut graphs are graphs whose adjacency matrix is singular with one-dimensional null space spanned by a vector with no zero entries. In a recent paper, Bašić, Fowler and Pisanski proved that the automorphism group of a nut graph has more orbits on the edge set than on the vertex set. They classified all orders for which a vertex-transitive nut graph with precisely two edge orbits exists, and conjectured that a nut graph with two vertex and three edge orbits exists for each non-prime order $n \ge 9$. Motivated by this conjecture, we introduce a very general construction that provides graphs with the desired symmetry properties, and we determine some sufficient spectral and structural conditions under which they are nut graphs. The construction yields infinite families of examples and confirms the above conjecture for all odd non-prime orders up to $2\,500$ and for at least $99.8$ percent of all odd non-prime orders up to a million. Finally, we present some additional interesting examples of nut graphs with two vertex and three edge orbits that do not arise from this construction.

math.CO↗
arXiv2026

On vertex-minimal simplicial maps to the sphere

For positive integers $n,d$, let $λ(n,d)$ be the minimal number of vertices of a triangulation of the $n$-sphere which admits a degree $d$ simplicial map onto the boundary of the $(n+1)$-simplex. We show that for $h=\lfloor\frac{n+1}2\rfloor$, the function $λ(n,d)^h$ has linear order of growth in $d$, answering a question of O. Musin. All triangulations we obtained are isomorphic to boundaries of convex polytopes in $\mathbb{R}^{n+1}$.

math.CO↗