Bulletin of the Belgian Mathematical Society - Simon Stevin

Counting Unlabeled Bipartite Graphs Using Polya's Theorem

Abdullah Atmaca and A. Yavuz Oruç

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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


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