On Counting Constructions and Isomorphism Classes of $I$-Graphs

TL;DR

This paper analyzes the asymptotic density of specific classes of $I$-graphs, including generalized Petersen graphs and connected graphs, using advanced number theory techniques.

Contribution

It provides precise asymptotic density results for $I$-graphs, especially regarding their classification as generalized Petersen graphs and their connectivity.

Findings

Quantifies the proportion of $I$-graphs that are generalized Petersen graphs.

Estimates the density of connected $I$-graphs.

Utilizes analytic number theory methods for asymptotic analysis.

Abstract

We prove a collection of asymptotic density results for several interesting classes of the $I$-graphs. Specifically, we quantify precisely the proportion of $I$-graphs that are generalised Petersen graphs as well as those that are connected. Our results rely on the estimation of sums over tuples satisfying various coprimality conditions along with other techniques from analytic number theory.