Graphs and quivers #
Graphs #
Graphs #
At a high level, a graph \( \graph{G} \) is a set of vertices \( \setSymbol{V} \), and a collection of edges \( \setSymbol{E} \).
These edges can be of different types, but fundamentally they each involve two vertices.
Directed graphs distinguish the two vertices in an edge. Such a directed edge is written \( \de{\vert{s}}{\vert{t}} \). The vertex \( \vert{s} \) is called th source and \( \vert{t} \) is called the target of the edge.
Undirected graphs do not make a distinction. Such undirected edges are written \( \ue{\vert{u}}{\vert{v}} \). The lack of distinction means that \( \ue{\vert{u}}{\vert{v}}\syntaxEqualSymbol \ue{\vert{v}}{\vert{u}} \).
A graph with self-loops allows the two vertices to be the same element from the set \( \setSymbol{V} \). A graph without self-loops requires the two vertices be distinct.
Labeled graphs #
Labeled graphs can attach labels to vertices and/or edges. These labels can be repeated, that is, shared between many vertices or edges.
Labeled edges are written \( \tde{\vert{s}}{\vert{t}}{\sym{l}} \), where \( \sym{l} \) is the label on the edge.