Enumerating Labeled Graphs that Realize a Fixed Degree Sequence

TL;DR

This paper presents a recurrence relation and recursive algorithm for counting labeled graphs that realize a fixed degree sequence, improving efficiency especially for regular graphs compared to generating function methods.

Contribution

It introduces a simple recurrence relation and an efficient recursive algorithm for exact enumeration of graphs with a given degree sequence, including regular graphs.

Findings

Exact count of labeled graphs for a fixed degree sequence via recurrence

Recursive algorithm with improved complexity for regular graphs

Better efficiency than generating function methods for certain cases

Abstract

A finite non-increasing sequence of positive integers $d = (d_1\geq \cdots\geq d_n)$ is called a degree sequence if there is a graph $G = (V,E)$ with $V = \{v_1,\ldots,v_n\}$ and $deg(v_i)=d_i$ for $i=1,\ldots,n$. In that case we say that the graph $G$ realizes the degree sequence $d$. We show that the exact number of labeled graphs that realize a fixed degree sequence satisfies a simple recurrence relation. Using this relation, we then obtain a recursive algorithm for the exact count. We also show that in the case of regular graphs the complexity of our algorithm is better than the complexity of the same enumeration that uses generating functions.