Graph

def. Graph. A Graph is defined as:

• Ordered pair $(V,E)$ where
  • $V$ is the set of all vertices
  • $E=\{x,y|x,y\in V\text{ and }x\ne y\}$

Types of Graphs §

• Variations on:
  • Directed or Undirected
  • Connected or Not Connected
  • Cyclical or Acyclical
• Common types:
  • Tree is a connected undirected acyclical graph
    • Forest is a set of trees
  • Directed Graph
    • Directed Acyclical Graph

Properties of Graphs §

• Degree of a vertex $v\in V$: number of edges that connect to $v$
• Loop: an edge that connects a vertex to itself
• Cycle: a path that starts and ends with the same vertex

Representation of Graphs in Memory §

1. Adjacency Matrix
   1. 2D table of all nodes: Store a 1 if the edge between two nodes exists, 0 otherwise. Example
2. Adjacency List
   1. Array of all vertices, which are also linked list that list all reachable neighbors. Example

• Complexities for the two types of representations:

Paths §

• $v_1\to v_2\to \cdots \to v_k$
• A Simple Path is one that does not repeat verticies

Special Variants §

• A Metric weighted graph is where the edge weights satisfy the triangle inequality; i.e. the vertices lie on a surface, and the edges are Euler distances of the verteces