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 all $a,b$ in $S$, $a...

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