The Parallel-Sequential Duality : Matrices and Graphs

TL;DR

This paper explores the duality between matrices and graphs, showing how reflexive directed graphs can be viewed as sequential programs that construct and are constructed by other objects, revealing new computational and structural insights.

Contribution

It introduces a novel perspective on mathematical objects as sequential constructors, establishing a duality between matrices and graphs with applications in memory computation and coding.

Findings

Reflexive directed graphs can be interpreted as programs that build and are built by other objects.

The duality leads to optimal memory computations and modular decompositions.

Reveals new dynamical phenomena in the structure of graphs and matrices.

Abstract

Usually, mathematical objects have highly parallel interpretations. In this paper, we consider them as sequential constructors of other objects. In particular, we prove that every reflexive directed graph can be interpreted as a program that builds another and is itself builded by another. That leads to some optimal memory computations, codings similar to modular decompositions and other strange dynamical phenomenons.