Remarks about the Arithmetic of Graphs

TL;DR

This paper extends classical arithmetic to graphs, creating a framework that incorporates topology, spectral theory, and number theory, with applications to primes and algebraic structures.

Contribution

It introduces a novel graph arithmetic framework that generalizes number systems and integrates topology, spectral theory, and number theory concepts.

Findings

Graphs form a semiring with arithmetic operations.

Most graphs are multiplicative primes in this framework.

The approach connects graph theory with algebraic and number-theoretic concepts.

Abstract

The arithmetic of N, Z, Q, R can be extended to a graph arithmetic where N is the semiring of finite simple graphs and where Z and Q are integral domains, culminating in a Banach algebra R. A single network completes to the Wiener algebra. We illustrate the compatibility with topology and spectral theory. Multiplicative linear functionals like Euler characteristic, the Poincare polynomial or the zeta functions can be extended naturally. These functionals can also help with number theoretical questions. The story of primes is a bit different as the integers are not a unique factorization domain, because there are many additive primes. Most graphs are multiplicative primes.