1.0 - Graphs

Graphs are comprised of vertices and edges.

• Vertices $\{A, B, C, D, E, F\}$
• Edges $\{(A, B), (A, D), (A, C), (B, D), (C, D), \\(E, D), (E, F), (D, F) \}$

Could be used to represent different types of information

• Geographical information (Vertices = cities, Edges = roads)
• Networks (Vertices = computers, Edges = cables)
• Brains (Vertices = neurons, Edges = synapses)
• Programs (Vertices = statements, Edges = control flow)

1.1 - Directed or Undirected Graphs

1.1.1 - Undirected Graphs

• Cannot have self-loops
• If $(u, v)\in E(G)$, $v$ is adjacent to u in G (and u is adjacent to v) - symmetric property
• Edge $(u,v)\in E(G)$ is incident on vertices $v, u$

1.1.2 - Directed Graphs

• Can have self-loops
• If $(u,v )\in E(G)$, $v$ is adjacent to $u$ - asymmetric property
• Edge $(u, v)\in E(G)$ is incident from (leaves) vertex $u$ and incident on (enters) vertex $v$

• The out-degree of a vertex $v$ is the number of edges leaving it
• The in-degree of a vertex v is the number of edges entering it
• The degree of a vertex is the sum of its in-degree and out-degree.
• Maximum number of edges $n^2$

1.2 - Weighted or Unweighted Graphs

1.2.1 - Unweighted Graphs

• A graph in which edges don’t have weights associated with them

1.2.2 - Weighted Graph

• A graph in which each edge has a weight associated with it.

1.3 - Paths

For a graph $G=(V, E)$

• A path of length $k$ from a vertex $v_0$ to a vertex $v_k$ is a sequence $\langle v_0, v_1, \cdots, v_k\rangle$ of vertices such that $(v_{i-1}, v_i)\in E$ for $i\in\{1, 2, \cdots, k\}$
  • That is, every two consecutive vertices form an edge that is in the graph.
• $u$ is reachable from $v$ if there is path from $v$ to $u$
• A path is simple if all of the vertices i nthe path are distinct
• A path forms a cycle if $v_0=v_k$ and $k>1$
• A cycle is simple if $v_1, \cdots, v_k$ are distinct, and all of its edges are distinct
• A graph with no simple cycles is acyclic

1.4 - Undirected Graphs - More Properties

• An undirected graph is connected if every vertex is reachable from all vertices

  • An undirected graph is connected if every vertex is reachable from all other vertices

    

  • An undirected graph is a forest if it is acyclic

    

  • An undirected graph is a tree if it is a forest (i.e. acyclic) with only one connected component.

    

What are the minimum and maximum number of edges in a:

• Tree with $n$ vertices? $(n-1, n-1)$
• Forest with $n$ vertices? $(0, n-1)$
  • Doesn’t have to be connected, so minimum is 0
• Connected graph with $n$ vertices? $(n-1, \frac{n(n-1)}{2})$

1.5 - Directed Graphs - More Properties

A directed graph $G=(V, E)$ is:

• Strongly connected if every two vertices are reachable from each other

1.6 - Graph Terminology

• Graph $G'=(V', E')$ is a sub-graph of $G=(V, E)$ when $V'\subseteq V$ and $E'\subseteq E$

• $G'$ is a spanning sub-graph of $G$ if the above, and $V'=V$ also holds.

• The sub-graph of $G=(V, E)$ that is induced by $V'$ is the graph $G'=(V', E')$ where:

  $$E'=\{(u, v)\in E\ :\ u \in V'\wedge v \in V'\}$$