
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 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 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 vertices.
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