Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.29486

Rooted spanning forests in wheel graphs: Fibonacci-Lucas formulas and extremal root configurations

Shunya Tamura·Yuuho Tanaka
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we study rooted spanning forests in the wheel graph $W_{N+1}$. For a root set $R$ consisting of rim vertices, we give an explicit formula, in terms of Fibonacci and Lucas numbers, for the number of rooted spanning forests in which each connected component contains exactly one vertex of $R$. When the central vertex is also included in the root set, we obtain a simple product formula involving only Fibonacci numbers. Furthermore, by using the correspondence between rooted spanning forests and vertex identification, we derive explicit formulas for the number of spanning trees of quotient graphs of wheel graphs obtained by identifying several vertices into one vertex. We then rewrite the obtained formulas combinatorially in terms of the numbers of matchings in path graphs and cycle graphs. In addition, for a fixed number $r$ of rim root vertices, we express the sum of the numbers of rooted spanning forests over all rim root sets of size $r$ as coefficients of generating functions, and we solve the extremal problem for the arrangement of roots.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Shunya Tamura, Yuuho Tanaka. 2026-09-04. Rooted spanning forests in wheel graphs: Fibonacci-Lucas formulas and extremal root configurations. https://arxiv.org/abs/2609.29486

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↗