Differential Calculus on Cayley Graphs

TL;DR

This paper develops a differential calculus framework on Cayley graphs within convergence spaces, extending classical calculus to algebraic structures like groups and Boolean algebras, with potential applications in algebra and graph theory.

Contribution

It introduces a novel differential calculus on Cayley graphs using convergence spaces, bridging graph theory and algebra with new calculus tools.

Findings

Established a differential calculus on groups via Cayley graphs

Derived a Boolean differential calculus satisfying linearity and Leibniz rule

Extended classical calculus concepts to convergence space representations of graphs

Abstract

We conservatively extend classical elementary differential calculus to the Cartesian closed category of convergence spaces. By specializing results about the convergence space representation of directed graphs, we use Cayley graphs to obtain a differential calculus on groups, from which we then extract a Boolean differential calculus, in which both linearity and the product rule, also called the Leibniz identity, are satisfied.