need a formula

jzellt · #1 $Old$ April 30th, 2008, 09:57 PM

3 Vertices = 3 possible edges
4 Vertices = 6 possible edges
5 Vertices = 10 possible edges
6 Vertices = 14 possible edges

Can anyone come up with a formula to calculate the number of edges with n vertices?

Soroban · #2 $Old$ April 30th, 2008, 10:12 PM

Hello, jzellt!

Quote:

$\begin{array}{ccc}3\text{ Vertices} &=& 3\text{ possible edges} \\ 4\text{ Vertices} &=& 6\text{ possible edges} \\ 5\text{ Vertices} &=& 10\text{ possible edges} \\ 6\text{ Vertices} &=& {\color{red}15}\text{ possible edges} \end{array}$

Can anyone come up with a formula for the number of edges with n vertices?

You might notice that we have Triangular Numbers . . .

. . . $\begin{array}{cc}\text{Vertices} & \text{Edges} \\ \hline 2 & 1 \\ 3 & 3 \\ 4 & 6 \\ 5 & 10 \\ 6 & 15 \\ \vdots & \vdots \end{array}$

Therefore: . $\boxed{E(n) \;=\;\frac{n(n-1)}{2}}$