Introduction to Graphs (For real this time)

• A graph is used to encode relationships between objects.
• The objects
  • Are represented as nodes or vertexes
  • Frequently represented as $V$, with $v\in V$ a single vertex.
  • Data is frequently stored at a vertex.
• The relationships
  • Are represented as edges
  • Frequently written as $E$, with $e \in E$ a single edge.
• Thus a graph $G = (V,E)$
• Some graph terminology
  • Most graphs are undirected
    • or if $e=(u,v), e \in E, u,v \in V$
    • then $ (v,u) \in E$
    • Or $(u,v) \in E \implies (v,u) \in E$
  • But can be directed.
    • Or $(u,v) \in E \mathrel{\rlap{\hskip .5em/}}\Longrightarrow (v,u) \in E$
  • Frequently all edges have the same (or no) weight.
  • But if the graph has weights associated with each edge it is a weighted graph.
  • A tree is a special case of a graph.
  • An acyclic graph has no cycles.
    • Trees usually have a starting point or a root
    • Trees are acyclic.
  • A special case of a graph is a Directed Acyclic Graph or DAG.
    • While we will not deal with these immediately.
    • We will encounter them.
    • A tree is a DAG
    • But there are other DAGS as well.
• Application of graphs. Read this (P 74)
  • Transportation , your GPSR
  • Communication Networks.
  • Informational Networks (web)
  • Social Networks (friends)
  • Dependency (make) (DAG)