IOI Training

1
IOI Training

• 11 June 2005

2
Graphs

• G (V,E) e E v V
• Show that if G is simple, then
• Show that there are 11 non-isomorphic simple
  graphs on 4 vertices.
• Let G be bipartite. Show that the vertices of G
  can be enumerated so that the adjacency matrix of
  G has the formwhere A21 is the transpose of
  A12.

3
Graphs

• A is the adjacency matrix of a graph G. M is the
  incidence matrix of G.
• If G is simple, the entries on the diagonals of
  both MM and A2 are the degrees of the vertices
  of G.
• Show that in any group of two or more people,
  there are always two with exactly the same number
  of friends inside the group.
• Show that the number of (vi, vj)-walks of length
  k in G is the (i, j)th entry of Ak.