Representing Graphs

• There are two natural ways to represent a graph.
• But first some terms.
  • A graph $G=(V,E)$ is dense if $|E| \approx |V|^2$
  • A graph $G=(V,E)$ is sparse if $|E| \approx |V|$
  • Why are these reasonable definitions?
    • For a fully connected graph, or complete graph, every vertex is connected to every other vertex.
      • Thus there are $|V|-1$ unique edges from the first vertex
      • And there are $|V|-2$ unique edges from the second vertex (don't count the edge from the first to the second)
      • ...
      • $\sum_{i=1}^{|V|-1}i$ edges.
      • So there are $O(n^2)$ edges in a complete graph.
    • For a tree on the other hand we have seen that $|V|=|E|-1$.
• We have two different forms of representing graphs in code.
  • An adjacency matrix is used to represent dense graphs
    • This is a two dimensional matrix where
      • the rows can represent connections from the first vertex u
      • the columns can represent connections to the second vertex v
      • The intersection represents the presence of an edge (u,v)
      • If the graph is weighted, it is customary to store the weight
      • If the graph is directed M[u,v] is not the same as M[v,u]
    • This requires $O(|V|^2)$ space
    • But checking for an edge is $O(1)$
    • And checking all adjacent edges is $O(|V|)$
  • An adjacency list is used to represent sparse graphs.
    • Generally this is an array of lists.
      • Each entry in the array represents adjacent vertexes to a given vertex.
      • This is usually in the form of a linked list.
    • In this case, it requires $(|V|+|E|)$ space (likely $2|E|$, but ...
    • It requires at most $O(|V|)$ to check for an edge, but this is probably much lower (since it is sparse)
    • It requires at most $O( \textrm{adj}(V))$ to find adjacencies.(just traverse the list)