On The Number Of Labeled Bipartite Graphs
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866911781851496448 |
|---|---|
| author | Atmaca, Abdullah Oruc, A. Yavuz |
| author_facet | Atmaca, Abdullah Oruc, A. Yavuz |
| contents | Let $I$ and $O$ denote two sets of vertices, where $I\cap O =\emptyset$, $|I| = n$, $|O| = r$, and $B_u(n,r)$ denote the set of unlabeled graphs whose edges connect vertices in $I$ and $O$. Recently, it was established in\cite{atmacaoruc2018} that the following two-sided equality holds, \begin{equation} \label{mainResult} \displaystyle \frac{\binom{r+2^{n}-1}{r}}{n!}\, \le\, |B_u(n,r)|\, \le\, 2\frac{\binom{r+2^{n}-1}{r}}{n!},\, n < r.\nonumber \end{equation} and exact formulas were provided in~\cite{atmaca2017size} for $|B_u(2,r)|$ and $|B_u(3,r)|.$ In this paper, these results are extended to various families of labeled bipartite graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_08053 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On The Number Of Labeled Bipartite Graphs Atmaca, Abdullah Oruc, A. Yavuz Combinatorics Let $I$ and $O$ denote two sets of vertices, where $I\cap O =\emptyset$, $|I| = n$, $|O| = r$, and $B_u(n,r)$ denote the set of unlabeled graphs whose edges connect vertices in $I$ and $O$. Recently, it was established in\cite{atmacaoruc2018} that the following two-sided equality holds, \begin{equation} \label{mainResult} \displaystyle \frac{\binom{r+2^{n}-1}{r}}{n!}\, \le\, |B_u(n,r)|\, \le\, 2\frac{\binom{r+2^{n}-1}{r}}{n!},\, n < r.\nonumber \end{equation} and exact formulas were provided in~\cite{atmaca2017size} for $|B_u(2,r)|$ and $|B_u(3,r)|.$ In this paper, these results are extended to various families of labeled bipartite graphs. |
| title | On The Number Of Labeled Bipartite Graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2402.08053 |