On some numerical characteristics of a bipartite graph

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 $g=⟨R_g\cup C_g,E_g⟩$ 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.