OUTLINE

Introduction
Definitions

Directed Graphs
Infinite Chains (Def. 1 & Def. 2)

When are Def. 1 and Def. 2 equivalent?

A fix with the axiom of countable choice

Directed Graphs in Applications

Softwares
Graph Neural Networks
Quantum Information

SUMMARY: We introduce two notions of an infinite chain in a directed graph, and we show when these two notions are equivalent. We then mention the use of directed graphs in applications, such as artificial intelligence and quantum information.

Introduction
Consider this diagram
$\begin{array}{ccccc} \bullet & \rightarrow & \bullet\\ \downarrow & & \downarrow\\ \bullet & \rightarrow & \bullet & \rightarrow & \bullet \end{array}$
which consists of vertices (the dots) and arrows between the vertices. That is an example of a directed graph.

Definitions
Directed Graph
A directed graph is a set $V$ with a binary relation $\rightarrow$. By a binary relation we mean a subset of the Cartesian product $V\times V$. A directed sub-graph of $V$ is a subset $S$ of $V$ with a binary relation $\rightarrow_{S}$ such that for a...