Question

find the maximum number of vertices in a connected graph having 17 edges

Answer 1

If the number of vertices is n
Then, maximum number of edge = $\binom{n}2 =\frac{n!}{2!(n-2)!}\\\\=\frac{n(n-1)}{2}$
⇒ n(n - 1)/2 = 17
⇒ n(n - 1) = 34
⇒ n² - n - 34 = 0
⇒ n = { 1 ± √(1 + 136)}/2
⇒ n = {1 ± √137}/2
⇒ n ≈ ( 1 ± 12)/2
⇒ n ≈ 6.5
Hence, maximum number of vertices = 6