Year: 2025
Type: preprint
Abstract: Let $\mathcal{T}_n$ be the set of all mappings $T:[n]\to[n]$, where $[n]=\{1,2,\ldots,n\}$. The corresponding graph $G_T$ of $T$, called a functional digraph, is a union of disjoint connected components. Each component is a directed cycle of rooted labeled trees. We assume that each $T\in\mathcal{T}_n$ is chosen uniformly at random from the set $\mathcal{T}_n$. The components and trees of $G_T$ are distinguished by their size. In this paper, we compute the limiting conditional probability ($n\to\infty$) that a vertex from the largest component of the random graph $G_T$, chosen uniformly at random from $[n]$, belongs to its $s$-th largest tree, where $s\ge 1$ is a fixed integer. This limit can be also viewed as an approximation of the probability that the $s$-th largest tree of $G_T$ is a subgraph of its largest component, which is a solution of a problem suggested by Mutafchiev and Finch (2024).
Source: arXiv (Cornell University)
Authors: Mutafchiev, Ljuben, Finch, Steven
Institution: American University in Bulgaria
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4417460347
Cites:
Cited by:
Related to: