Arithmetical Semirings

TL;DR

This paper investigates the properties of connected graphs that are prime with respect to the Cartesian product, providing asymptotic bounds and expansions, and introduces a novel approach inspired by arithmetical semigroup theory.

Contribution

It introduces a new method based on arithmetical semigroup theory to analyze the asymptotic behavior of connected, Cartesian prime graphs.

Findings

Almost all large graphs are connected and Cartesian prime.

A full asymptotic expansion for graphs with even vertices is derived.

The method connects graph enumeration with arithmetical semigroup theory.

Abstract

We study the number of connected graphs with $n$ vertices that cannot be written as the cartesian product of two graphs with fewer vertices. We give an upper bound which implies that for large $n$ almost all graphs are both connected and cartesian prime. For graphs with an even number of vertices, a full asymptotic expansion is obtained. Our method, inspired by Knopfmacher's theory of arithmetical semigroups, is based on reduction to Wright's asymptotic expansion for the number of connected graphs with $n$ vertices.