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How many edges must a graph with N vertices have in order to guarantee that it is connected? 

here on stack overflow they said (n-1)*(n-2)/2 +1

but i have doubt in it suppose we consider graph on 10 vertices.

here we need just 21 edges for n=10

but according to given formula it is 9*8/2  + 1 = 37.

so what is correct?

in Graph Theory by (13 points) | 9 views
With 21 edges and 10 vertices, can't we have a disconnected graph?
If yes, it mean we can't guarantee graph is connected with 10 vertices and 21 edges.
How can we have disconnected graph (simple graph)

can you show , how can i add 21st edge in above graph such that after addition graph will be still disconnected and simple?
In that 10 vertices group... Make it complete graph..

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