Prove that for n ⩾ 2 there are exactly two n-vertex graphs with n − 1...

Prove that for n ⩾ 2 there are exactly two n-vertex graphs with n − 1 distinct degrees (up to isomorphism). The other answers on the website are incorrect.

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Colby Messinger answered 1 year ago

prove 2 is a factor of (n+1)(n+2) for all positive integers

Prove that 1^3 + 2^3 + · · · + n^3 = (1 + 2 +...

Prove that 1^3 + 2^3 + · · · + n^3 = (1 + 2 + · · · + n)^2 for every n ∈ N. That is, the sum of the first n perfect cubes is the square of the sum of the first n natural numbers. (As a student, I found it very surprising that the sum of the first n perfect cubes was always a perfect square at all.)

Prove that for all integers n ≥ 2, the number p(n) − p(n − 1) is...

Prove that for all integers n ≥ 2, the number p(n) − p(n − 1) is equal to the number of partitions of n in which the two largest parts are equal.

By induction: 1. Prove that Σni=1(2i − 1) = n2 2. Prove thatΣni=1 i2 = n(n+1)(2n+1)...

By induction: 1. Prove that Σni=1(2i − 1) = n2 2. Prove thatΣni=1 i2 = n(n+1)(2n+1) / 6 .

Prove the following by induction: 2 + 4 + 6 + …+ 2n = n(n+1) for...

Prove the following by induction: 2 + 4 + 6 + …+ 2n = n(n+1) for all integers n Show all work

Let ?=2^(2^?)+1 be a prime that n>1 1. Show that ? ≡ 2(mod5) 2. Prove that...

Let ?=2^(2^?)+1 be a prime that n>1 1. Show that ? ≡ 2(mod5) 2. Prove that 5 is a primitive root modulo ?

Prove that if n is an integer and n^2 is even the n is even.

1.Prove that{2k+1:k∈N}∩{2k2 :k∈N}=∅. 2.Give two examples of ordered sets where the meaning of ” ≤ ”...

1.Prove that{2k+1:k∈N}∩{2k2 :k∈N}=∅. 2.Give two examples of ordered sets where the meaning of ” ≤ ” is not the same as the one used with the set of real numbers R.

Prove that the number of partitions of n into parts of size 1 and 2 is...

Prove that the number of partitions of n into parts of size 1 and 2 is equal to the number of partitions of n + 3 into exactly two distinct parts

Discrete Mathematics A tree contains 1 vertex of degree 2, 1 vertex of degree 3, 1...

Discrete Mathematics A tree contains 1 vertex of degree 2, 1 vertex of degree 3, 1 vertex of degree 4, 11 leaves and the remaining vertices have degree 3. Find the total number of vertices. Sketch two non-isomorphic trees statisfying the above mentioned conditions.

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