On the Enumeration of Interval Graphs

TL;DR

This paper investigates the enumeration of interval graphs, providing bounds on their count and analyzing the convergence properties of their generating functions.

Contribution

It establishes bounds for the number of interval graphs and analyzes the radii of convergence of their generating functions, answering a question posed by Hanlon.

Findings

The ordinary generating function for interval graphs has radius of convergence zero.

The exponential generating function for interval graphs has radius of convergence at least 1/2.

Provides bounds for the number of interval graphs on n vertices.

Abstract

We present upper and lower bounds for the number $i_n$ of interval graphs on $n$ vertices. Answering a question posed by Hanlon, we show that the ordinary generating function $I(x) = \sum_{n\ge 0} i_n\,x^n$ for the number $i_n$ of $n$-vertex interval graphs has radius of convergence zero. We also show that the exponential generating function $J(x) = \sum_{n\ge 0} i_n\,x^n/n!$ has radius of convergence at least $1/2$.