On The Number Of Unlabeled Bipartite Graphs

TL;DR

This paper derives an asymptotic formula for counting unlabeled bipartite graphs by establishing a two-sided equality using Polya's Counting Theorem, solving a problem posed in 1973.

Contribution

It provides the first asymptotic formula for the number of unlabeled bipartite graphs, resolving a long-standing open problem from 1973.

Findings

Established a two-sided asymptotic formula using Polya's Counting Theorem.

Connected the enumeration of binary matrices to unlabeled bipartite graphs.

Solved a problem that remained open for nearly 50 years.

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.