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Random Graphs

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Title: Random Graphs


1
Random Graphs
  • Ben Byer
  • Ara Hayrapetyan
  • Jeff Philips

2
Graph background
  • What is a graph?
  • Collection of vertices and edges, or points and
    adjacencies
  • We will discuss a graph G

3
Common permutations of graphs
  • Subgraph vertices and edges removed
  • Induced subgraph only vertices removed

4
More permutations
  • Complement graph invert the existence of edges

5
Notation
  • n(g) number of vertices in graph g
  • e(g) number of edges in graph g

6
Common functions of graphs
  • girth - g(G) - the length of the shortest simple
    cycle in a graph
  • chromatic number - ?(G) - the smallest number of
    colors needed to color every vertex, so that no
    two adjacent vertices share the same color.
  • diameter - d(G) - the length of the longest path
    in the set of shortest paths between every pair
    in a graph

7
More functions
  • clique number - ?(G) - the size (in vertices) of
    the largest clique in a graph
  • independence number - ?(G) - the size(in
    vertices) of the largest set of non-connected
    nodes in a graph
  • k-connected - two vertices are k-connected if
    they are mutually adjacent to k other vertices.
    (A graph is k-connected if every pair of vertices
    is k-connected.)

8
n(G)
  • n(G) ? ?(G) ?(G)
  • Proof
  • graph will have ?(G) independent subgraphs, one
    per color
  • each of these independent subgraphs is no larger
    than ?(G) (a given colors set is independent by
    definition)

9
What is a random graph?
  • Vertex definition - n vertices, p probability of
    each edge
  • Edge definition - n vertices, m edges
  • Theyre equivalent - we will use vertex
    definition
  • G(n,p) set of all random graphs with parameters
    n and p

10
Properties of all graphs
  • Isomorphism Two graphs are isomorphic if a
    mapping exists between the vertices of the graph,
    such that the edges are preserved
  • Property (on a graph) A quality of a graph that
    is true for all isomorphic graphs

11
Why random graphs?
  • Used to prove existence of graphs with specific
    properties
  • Example Erdos
  • Exist graphs with arbitrarily large ?(G) and g(G)

12
Almost all graphs
  • Property P holds for almost all graphs if the
    probability that it holds for some G tends to 1,
    as n??

13
Application melting point
  • Model of matter as a random, connected graph
  • Largely connected solid as bonds break, state
    changes
  • Actual structure does not matter, only rough
    arrangement and connectedness
  • Exist thresholds, analogous to edge
    probabilities, at which these states change
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