Let us say Simple graph has V vertices and E edges.
Now i am trying to find relation between E and V.
If it is a complete graph ,then E=V(V-1)/2,
=> E=$c.V^2$
=> E=$O(V^2)$
=> taking log of both sides.
=> $\log E$=$O(\log V)$------------------ (1)
Now in case we are fining spanning tree from graph ,we now the number of edges will be minimum V-1 in graph.So i can say
V-1<=E
V=O(E)
=> taking log of both sides.
=>$\log V$=$O(\log E)$------------------ (2)
So ,both the equations are coming valid. So is can if i say,combing both the equation i can say.
$\log E$ =theta$(\log V)$.------------------(3)
Can someone confirm if the 1 ,2 and the result of them i.e the 3rd equation is valid ?