An introduction to Graphs

• A graph G=<V,E> is a set of vertices (V) and a set of edges (E).
• An edge e=(u,v), e∈E u∈V, v∈ V, connects two verticies.
• If e =(u,v) then u and v are adjacent.
• Edges can be directed e=(u,v) does not imply e'=(v,u)
• Or undirected e=(u,v) = (v,u)
• If the edges are directed, the graph is called a directed graph or a digraph
• Edges may have an associated weight, nor not (all edge costs are the same).
• While an edge e=(u,u) is permitted, we will usually consider graphs without such edges.
• |E| is the number of edges in a graph.
• |V| is the number of verticies in a graph.
• If every pair of verticies has a connecting edge, the graphs complete.
• A graph where |E| ∈ O(|V|²) is dense
• A graph where |E| ≈ |V| is sparse.
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• two verticies u, and v are connected if e=(u,v) exists.
• a path is a series of connected verticies.
• a cycle is a path that starts and ends at the same vertex.
• Graph Representation
  • A dense graph is represented by an adjacency matrix.
  • A sparse graph is represented by an adjacency list.
  • We will frequently want to

    for each  vertex v in G
        for each edge e incident to v
        or
        for each vertex u adjacent to v