4 - Operator Graph Theory: The Mathematics of Finite Networks

Summary

Over the course of three centuries, the field of graph theory has matured from its initial conception as an abstract model for solving a rather specific mathematical problem into a powerful vessel for describing countless real-world phenomena, with applications now reaching far beyond applied mathematics. However, its static formalisation puts a number of limitations in place which hamper an advantageous utilisation of graph-theoretical concepts in circumstances which require a more dynamical perspective. Is it possible to overcome these limitations by challenging the classical notion of a graph? In this chapter, we will propose exactly such a challenge by considering a graph’s nodes and their relations as the result of operations performed on a set of suitable objects. This subtle yet consequential change in the conception of a graph not only delivers a more dynamic vantage point, but eventually generalises the very notion of a graph by structurally equating it with an abstract algebra. This chapter will introduce the basic notions and formalisations of an operator graph-theoretical framework and candidly argue for its potential merits and usefulness.