1
$\begingroup$

If every pair of vertices in a graph has at most $k$ common neighbors and at most $k$ non common neighbors then what is the bound on the number of vertices ? The graph has connected components .

The answer is at most $(2k+1)\cdot k$.

But I don't understand. Can someone explain ..

Since each vertex has $k$ common neighbors and $k$ non common neighbors it should have at least ($k$(neighbors)+$k$(non common neighbors)+$1$(referring vertex)) $2k+1$ vertices . But how it is at most $(2k+1)\cdot k$?

For Example k=1 we have max vertices =3 .
Let the vertices be u ,v and w .
E={(u,v),(v,w)}//Edge Set
For pair (u,v): 0 common neighbor and 1 non common neighbor (because of edge v-w )
For pair (v,w): 0 common neighbor and 1 non common neighbor (because of edge u-v )
For pair (u,w): 1 common neighbor (because of edges u-v and v-w) and 0 non common neighbors

  • 0
    I understand that a "common neighbor" of two vertices $u$ and $v$ is a vertex which is adjacent to both $u$ and $v.$ I'm not sure what a "non common neighbor" is. Is it a vertex which is *not* a common neighbor, meaning that it's adjacent to at most one of them? Or is it a vertex which is adjacent to neither of them? – 2017-02-21
  • 0
    It means the pair has k non common neighbors of (u,v) – 2017-02-21
  • 1
    I asked you what it means to be a "non common neighbor". Your reply answers some other question. – 2017-02-21
  • 1
    Is it a vertex which is not a common neighbor, meaning that it's adjacent to at most one of them? Yes .. – 2017-02-21

1 Answers 1

0

The graph has connected components

So in one component following property holds: Every vertex can connect to k vertices and not connect to k vertices . So the total number of vertices is 2k+1 .
If u add an extra vertex in this graph it might violate the rule :
every pair of vertices in a graph has at most k common neighbors and at most k non common neighbors
If one component can have 2k+1 vertices then k components can have atmost (2k+1).k vertices .