Mathematics,  Premium

Two Notions of an Infinite Chain in a Directed Graph


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

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.

Consider this diagram
\bullet & \rightarrow & \bullet\\
\downarrow & & \downarrow\\
\bullet & \rightarrow & \bullet & \rightarrow & \bullet
which consists of vertices (the dots) and arrows between the vertices. That is an example of a directed graph.

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...

You are unauthorized to view this page.