Canonical binary matrices related to bipartite graphs

TL;DR

This paper introduces canonical binary matrices to uniquely represent bipartite graphs, providing a necessary and sufficient condition for their canonical form, which aids in counting and generating all non-isomorphic bipartite graphs.

Contribution

It defines canonical binary matrices for bipartite graphs and establishes a criterion for their canonical form, enabling enumeration of all non-isomorphic bipartite graphs.

Findings

Defined canonical binary matrices for bipartite graphs

Established a necessary and sufficient condition for canonical matrices

Proposed a recursive algorithm for enumeration

Abstract

The current paper is dedicated to the problem of finding the number of mutually non isomorphic bipartite graphs of the type $g=\langle R_g ,C_g ,E_g \rangle$ at given $n=|R_g |$ and $m=|C_g |$, where $R_g$ and $C_g$ are the two disjoint parts of the vertices of the graphs $g$, and $E_g$ is the set of edges, $Eg \subseteq R_g \times C_g$. For this purpose, the concept of canonical binary matrix is introduced. The different canonical matrices unambiguously describe the different with exactness up to isomorphism bipartite graphs. We have found a necessary and sufficient condition an arbitrary matrix to be canonical. This condition could be the base for realizing recursive algorithm for finding all $n \times m$ canonical binary matrices and consequently for finding all with exactness up to isomorphism binary matrices with cardinality of each part equal to $n$ and $m$.