Transcript Document 7599699

IOI Training
11 June 2005
Graphs
 G = (V,E); e = |E|; v = |V|
v
1. Show that if G is simple, then e   
 2
2. Show that there are 11 non-isomorphic simple graphs
on 4 vertices.
3. Let G be bipartite. Show that the vertices of G can be
enumerated so that the adjacency matrix of G has the
form
 0 A12
 A21 0 


where A21 is the transpose of A12.
Graphs
 A is the adjacency matrix of a graph G. M is the
incidence matrix of G.
4. If G is simple, the entries on the diagonals of both MM’
and A2 are the degrees of the vertices of G.
5. Show that in any group of two or more people, there
are always two with exactly the same number of friends
inside the group.
6. Show that the number of (vi, vj)-walks of length k in G is
the (i, j)th entry of Ak.