On some numerical characteristics of a bipartite graph

TL;DR

This paper introduces numerical characteristics based on an equivalence relation in bipartite graphs, providing a combinatorial identity relating these characteristics to the structure of all such graphs.

Contribution

It defines new numerical characteristics for bipartite graphs and proves a combinatorial identity connecting these to the graph's structure.

Findings

Defined equivalence-based numerical characteristics.

Formulated and proved a related combinatorial identity.

Applied to the set of all bipartite graphs with given vertex and edge counts.

Abstract

The paper consider an equivalence relation in the set of vertices of a bipartite graph. Some numerical characteristics showing the cardinality of equivalence classes are introduced. A combinatorial identity that is in relationship to these characteristics of the set of all bipartite graphs of the type $g=\langle R_g \cup C_g, E_g \rangle$ is formulated and proved, where $V=R_g \cup C_g$ is the set of vertices, $E_g$ is the set of edges of the graph $g$, $ |R_g |=m\ge 1$, $|C_g |= n\ge 1$, $|E_g |=k\ge 0$, $m,n$ and $k$ are integers.