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I have doubt regarding the following:-(n=number of vertices)

1) Minimum number of edges for a graph to be connected = n-1

2) maximum number of edges for a graph to be disconnected or to stay disconnected(not able to form correct wording. Please correct me here. I mean there is disconnected graph. I will keep on adding edges till it remain disconnected. I am thinking about how many maximum edges be there and still remain disconnected) = $ \binom{n-1}{2}$

My approach for (2)

Say graph is disconnected and there be 2 components. Then number of edges = $\binom{n_1}{2}+\binom{n_2}{2}, n_1, n_2$ are vertices in 1st and 2nd component respectively($n_1+n_2=n$). For edges to be maximum, gap between n1 and n2 has to be maximum for its sum constant. So n1 = 1 and n2 = n-1 or vice-versa. So in this case, the num of edges become $\binom{n-1}{2}$

Please correct me if i am wrong above both in (1) and (2)

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  • $\begingroup$ I guess $1)$ is true and it have special name tree. $\endgroup$ – emonHR Nov 23 '19 at 18:46
  • $\begingroup$ sir can u please have a look at this now? $\endgroup$ – Nascimento de Cos Nov 23 '19 at 19:58
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$1)$ is true and it has special name tree.
$2)$ I think number of edges may vary from graph to graph. Since you consider the graph is not connected it has at least two components. Even if it has more than $2$ components, you can think about it as having $2$ "pieces", not necessarily connected.

Let $k$ and $n-k$ be the number of vertices in the two pieces. Then, each vertex in the first piece has degree at most $k-1$, therefore the number of edges in the first component is at most $\frac{k(k-1)}{2}$, while the number of edges in the second component is at most $\frac{(n-k)(n-k-1)}{2}$.

To finish the problem, just prove that for $1 \leq k \leq k-1$ we have $$\frac{k(k-1)}{2}+ \frac{(n-k)(n-k-1)}{2} \leq \frac{(n-1)(n-2)}{2}$$

You can also prove that you only get equality for $k=1$ or $k=n-1$.

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  • $\begingroup$ sir can u please explain how u got that last inequality? $\endgroup$ – Nascimento de Cos Nov 24 '19 at 7:30
  • $\begingroup$ It's from your assumption $ \binom{n-1}{2}$ which bound all possible. $\endgroup$ – emonHR Nov 24 '19 at 7:56
  • $\begingroup$ Got it sir. Thank you ...... $\endgroup$ – Nascimento de Cos Nov 24 '19 at 7:57
  • $\begingroup$ I am glad to help you $\endgroup$ – emonHR Nov 24 '19 at 7:59

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