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Arithmetical Semirings

Marco Aldi

TL;DR

An upper bound is given which implies that for large $n$ almost all graphs are both connected and cartesian prime, and that for graphs with an even number of vertices, a full asymptotic expansion is obtained.

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.

Arithmetical Semirings

TL;DR

An upper bound is given which implies that for large almost all graphs are both connected and cartesian prime, and that for graphs with an even number of vertices, a full asymptotic expansion is obtained.

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.

Paper Structure

This paper contains 4 sections, 9 theorems, 88 equations.

Table of Contents

  1. introduction
  2. Arithmetical Semirings
  3. Asymptotic distribution of primes
  4. submultiplicative functions

Key Result

Lemma 21

Let $({\mathbb S},+,\Box,\partial)$ be an arithmetical semiring and suppose that there exist $\alpha,\beta,\gamma,a,b\in {\mathbb R}$ such that $\alpha>0$, $a>b>0$ and Then axioms ${\mathcal{G}}_1^+$ and ${\mathcal{G}}_2^+$ are satisfied. Moreover, if $B$ is the integer sequence defined recursively for all positive integers by the formula then for each integer $R>1$. In particular, axiom ${\mat

Theorems & Definitions (39)

  • Definition 1
  • Definition 2
  • Remark 3
  • Example 4
  • Definition 5
  • Definition 6
  • Remark 7
  • Example 8
  • Remark 9
  • Example 10
  • ...and 29 more