Graphs

• Definitions
  • A graph G is a set of vertices V and edges E
  • The vertices represent some item or thing
  • The edges represent a connection between them.
  • 
  • The terms edges and arcs are interchangeable
  • The terms nodes and vertices are interchangeable
  • Let e = (u,v), e in E. e is an edge from u to v, u,v IN V
  • If e =(u,v) in E => exists e' in E, e' = (v,u) then G is a undirected graph
  • If e=(u,v) in E !=> exists e' in E, e' = (v,u) then G is a directed graph.
  • Two vertices u,v in V are adjacent if exists e in E such that e=(u,v)
  • an edge e in E is incident on a vertex v in V if v is an endpoint of e
  • The degree of a vertex v, deg(v), is the number of incident edges.
    • deg(Erie) = 4, deg(Albion) = 1
  • In a digraph the degree in is the number of incoming edges and the degree out is the number of outgoing edges from a vertex v in V.
  • Edges can have an associated cost, in that case the graph is a weighted graph
  • Or edges can have no associated cost, in that the graph is a unweighted graph
  • A graph where |E| is less than |V²| is said to be sparse
  • A graph where |E| is close to |V²| is said to be dense
  • A graph G where for all u,v in V, exists e=(u,v) in E, G is fully connected
  • Most graphs don't allow e in E and u in V e = (u,u)
  • Most times if e = (u,v) and e' = (u,v) then e=e'
  • Prop 1: If G =(V,E) is a graph, |E| = m, then

    Each edge for (u,v) , it is counted once for u and once for v in the sum. Therefore the sum is twice the number of edges.

  • Prop 2: Let G=(V,E), |V| = n, |E| = m.

    If G is undirected
          m <= n(n-1)/2
    
    If G is directed
          m <= n(n-1)
    

  • A path from u to v = (e₁,e₂, ... e_n) where e₁ = (u, a), e₂ = (a,b), ... and e_n = (x,v)
  • A cycle is a path from u to u
  • A simple cycle contains each vertex once, except for the first and last.
  • If G=(V,E) and G' = (V',E') such that V' is a subset of V, and E' is a subset of E, then G' is a subgraph of G
  • if G=(V,E) and G' = (V,E') then G' is a spanning subgraph of G.