Question

please use contradiction

Prove the number of vertices of degree 1 in a tree must be greater than or equal to the maximum degree in the tree.

Answer #1

Prove the number of vertices of degree 1 in an tree must be
greater than or equal to the maximum degree in the tree.
(Try either Contradiction or Direct Proof)

Use proof by contradiction to prove that if T is a tree, then
every edge of T is a bridge.

Let G be an undirected graph with n vertices and m edges. Use a
contradiction argument to prove that if m<n−1, then G is not
connected

In lecture, we proved that any tree with n vertices must have n
− 1 edges. Here, you will prove the converse of this statement.
Prove that if G = (V, E) is a connected graph such that |E| =
|V| − 1, then G is a tree.

Graph Theory
.
While it has been proved that any tree with n vertices must have
n − 1 edges. Here, you will prove the converse of this statement.
Prove that if G = (V, E) is a connected graph such that |E| = |V |
− 1, then G is a tree.

10. (a) Prove by contradiction that the sum of an irrational
number and a rational number must be irrational. (b) Prove that if
x is irrational, then −x is irrational. (c) Disprove: The sum of
any two positive irrational numbers is irrational

please solve it step by step. thanks
Prove that every connected graph with n vertices has at least
n-1 edges. (HINT: use induction on the number of vertices
n)

Use proof by induction to prove that every connected planar
graph with less than 12 vertices has a vertex of degree at most
4.

Question 38
A simple connected graph with 7 vertices has 3 vertices of
degree 1, 3 vertices of degree 2 and 1 vertex of degree 3. How many
edges does the graph have?
Question 29
Use two of the following sets for each part below. Let X = {a,
b, c}, Y = {1, 2, 3, 4} and Z = {s, t}. a) Using ordered pairs
define a function that is one-to-one but not onto. b) Using ordered
pairs define...

(i) Use the Intermediate Value Theorem to prove that there is a
number c such that 0 < c < 1 and cos (sqrt c) = e^c- 2.
(ii) Let f be any continuous function with domain [0; 1] such
that 0smaller than and equal to f(x) smaller than and equal to 1
for all x in the domain. Use the Intermediate Value Theorem to
explain why there must be a number c in [0; 1] such that f(c)
=c

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