A note on observables for counting trails and paths in graphs

TL;DR

This paper shows that counting trails and paths in graphs can be achieved through quantum mechanical observables, linking graph theory problems with concepts from quantum physics and emergent geometry.

Contribution

It introduces a novel approach to compute graph path counts using quantum observables, connecting graph theory with physical theories of gravity.

Findings

Counting trails and paths as expectation values of quantum observables

Observables are related to background independent theories of gravity

Highlights the computational complexity in physical formalism

Abstract

We point out that the total number of trails and the total number of paths of given length, between two vertices of a simple undirected graph, are obtained as expectation values of specifically engineered quantum mechanical observables. Such observables are contextual with some background independent theories of gravity and emergent geometry. Thus, we point out yet another situation in which the mathematical formalism of a physical theory has some computational aspects involving intractable problems.