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december 2018 Counting Unlabeled Bipartite Graphs Using Polya's Theorem
Abdullah Atmaca, A. Yavuz Oruç
Bull. Belg. Math. Soc. Simon Stevin 25(5): 699-715 (december 2018). DOI: 10.36045/bbms/1547780430

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}

Citation

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Abdullah Atmaca. A. Yavuz Oruç. "Counting Unlabeled Bipartite Graphs Using Polya's Theorem." Bull. Belg. Math. Soc. Simon Stevin 25 (5) 699 - 715, december 2018. https://doi.org/10.36045/bbms/1547780430

Information

Published: december 2018
First available in Project Euclid: 18 January 2019

zbMATH: 07038547
MathSciNet: MR3901841
Digital Object Identifier: 10.36045/bbms/1547780430

Subjects:
Primary: 05A16, 05C30

Rights: Copyright © 2018 The Belgian Mathematical Society

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Vol.25 • No. 5 • december 2018
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