Counting Connected Graphs without Overlapping Cycles

TL;DR

This paper develops a mathematical framework to count simple connected graphs with arbitrary cycles that do not share nodes or edges, using advanced combinatorial and recursive methods.

Contribution

It introduces a novel counting series for such graphs, combining planted graph series, enriched trees, and a generalized Otter's formula.

Findings

Derived the counting series for graphs with non-overlapping cycles.

Established recursive methods for enumerating these graphs.

Connected the counting of rooted and unrooted graphs through a generalized formula.

Abstract

The simple connected graphs may be classified by their cycle composition (number and lengths of cycles). This work derives the counting series of the simple connected graphs that have cycles of unrestricted number and length, but no overlapping cycles. Cycle pairs of these graphs of interest must not have common nodes or edges. The recipe of counting these graphs is based on the counting series of the associated planted graphs, multisets of planted graphs, a recursive synthesis of enriched trees, and a generalized Otter's formula that maps the underlying rooted block graphs to the underlying block graphs.