Enumeration of graphs with given weighted number of connected components

TL;DR

This paper develops a generating function framework to count graphs with specific properties and a weighted number of connected components, including applications to bipartite graphs with fixed parameters.

Contribution

It introduces a new generating function approach for enumerating graphs with prescribed weighted connected components and applies it to bipartite graph enumeration.

Findings

Derived a generating function for graphs with given properties and weighted connected components.

Provided a specific generating function for bipartite graphs with fixed order, size, and components.

Abstract

We give a generating function for the number of graphs with given numerical properties and prescribed weighted number of connected components. As an application, we give a generating function for the number of bipartite graphs of given order, size and number of connected components.