## Bulletin of the Belgian Mathematical Society - Simon Stevin

- Bull. Belg. Math. Soc. Simon Stevin
- Volume 25, Number 5 (2018), 699-715.

### Counting Unlabeled Bipartite Graphs Using Polya's Theorem

Abdullah Atmaca and A. Yavuz Oruç

#### Abstract

This paper solves a problem that was stated by M. A. Harrison in 1973. The problem has remained open since then, and it is concerned with counting equivalence classes of $n\times r$ binary matrices under row and column permutations. 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 $O$. Harrison established that the number of equivalence classes of $n\times r$ binary matrices is equal to the number of unlabeled graphs in $B_u(n,r).$ He also computed the number of such matrices (hence such graphs) for small values of $n$ and $r$ without providing an asymptotic formula for $|B_u(n,r)|.$ Here, such an asymptotic formula is provided by proving the following two-sided inequality using Polya's Counting Theorem. \begin{equation} \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. \end{equation}

#### Article information

**Source**

Bull. Belg. Math. Soc. Simon Stevin, Volume 25, Number 5 (2018), 699-715.

**Dates**

First available in Project Euclid: 18 January 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1547780430

**Digital Object Identifier**

doi:10.36045/bbms/1547780430

**Mathematical Reviews number (MathSciNet)**

MR3901841

**Zentralblatt MATH identifier**

07038547

**Subjects**

Primary: 05C30: Enumeration in graph theory 05A16: Asymptotic enumeration

**Keywords**

Bipartite graphs cycle index polynomial graph enumeration Polya's counting theorem

#### Citation

Atmaca, Abdullah; Oruç, A. Yavuz. Counting Unlabeled Bipartite Graphs Using Polya's Theorem. Bull. Belg. Math. Soc. Simon Stevin 25 (2018), no. 5, 699--715. doi:10.36045/bbms/1547780430. https://projecteuclid.org/euclid.bbms/1547780430