%0 Generic %1 atmaca2017number %A Atmaca, Abdullah %A Oruc, A. Yavuz %D 2017 %K %T On The Number Of Unlabeled Bipartite Graphs %U http://arxiv.org/abs/1705.01800 %X This paper solves a problem that was stated by M. A. Harrison in 1973~harrison1973number. This problem, that has remained open since then is concerned with counting equivalence classes of $nr$ binary matrices under row and column permutations. Let $I$ and $O$ denote two sets of vertices, where $IO =\Phi$, $|I| = n$, $|O| = r$, 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 $nr$ 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 $|B_u(n,r)|.$ Here, such an asymptotic formula is provided by proving the following two-sided equality using Polya's Counting Theorem.

@misc{atmaca2017number, abstract = {This paper solves a problem that was stated by M. A. Harrison in 1973~\cite{harrison1973number}. This problem, that has remained open since then 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 $I\cap O =\Phi$, $|I| = n$, $|O| = r$, 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 $|B_u(n,r)|.$ Here, such an asymptotic formula is provided by proving the following two-sided equality using Polya's Counting Theorem.}, added-at = {2018-11-13T04:31:43.000+0100}, author = {Atmaca, Abdullah and Oruc, A. Yavuz}, biburl = {https://www.bibsonomy.org/bibtex/21f07955a488c434ddc28b44f25060689/shoaib-santo}, interhash = {4bd621fb280a25ebc0de2a726cf3dcea}, intrahash = {1f07955a488c434ddc28b44f25060689}, keywords = {}, note = {cite arxiv:1705.01800}, timestamp = {2018-11-13T04:31:43.000+0100}, title = {On The Number Of Unlabeled Bipartite Graphs}, url = {http://arxiv.org/abs/1705.01800}, year = 2017 }