Question

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 a function that is onto but not one-to-one.

Question 3

Consider the following statement: For all integers n, if n is odd then n is odd. 3 Prove the statement by Contraposition. Clearly indicate what you would “suppose” and what you would “show” to prove the statement.

Question 46 What is the second term of (2x^2 ‑ 3y)^6? Your answer should be in the form (sign)(numerical factors)(actual powers of variables).

Answer #1

Graph Theory.
A simple graph G with 7 vertices and 10 edges has the
following properties: G has six vertices of degree
a and one vertex of degree b. Find a and
b, and draw the graph.
Show all work.

Let G be a simple graph in which all vertices have degree four.
Prove that it is possible to color the edges of G orange or blue so
that each vertex is adjacent to two orange edges and two blue
edges.
Hint: The graph G has a closed Eulerian walk. Walk along it and
color the edges alternately orange and blue.

Prove that if G is a connected graph with exactly 4 vertices of
odd degree, there exist two trails in G such that each edge is in
exactly one trail. Find a graph with 4 vertices of odd degree
that’s not connected for which this isn’t true.

Suppose that a connected planar graph has eight vertices each of
degree 3 then how many regions does it have?And suppose that a
polyhedron has 12 triangular faces then determine the number of
edges and vertices.

6. If
a graph G has n vertices, all of which but one have odd degree, how
many
vertices of odd degree are there in G, the complement of G?
7.
Showthatacompletegraphwithmedgeshas(1+8m)/2vertices.

Show that if G is connected with n ≥ 2 vertices and n − 1 edges
that G contains a vertex of degree 1.
Hint: use the fact that deg(v1) + ... + deg(vn) = 2e

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)

You are given a directed acyclic graph G(V,E), where each vertex
v that has in-degree 0 has a value value(v) associated with it. For
every other vertex u in V, define Pred(u) to be the set of vertices
that have incoming edges to u. We now define value(u) = ?v∈P red(u)
value(v). Design an O(n + m) time algorithm to compute value(u) for
all vertices u where n denotes the number of vertices and m denotes
the number of edges...

Let G be a connected simple graph with n vertices and m edges.
Prove that G contains at least m−n+ 1 different subgraphs
which are polygons (=circuits). Note: Different polygons
can have edges in common. For instance, a square with a diagonal
edge has three different polygons (the square and two different
triangles) even though every pair of polygons have at least one
edge in common.

Prove that a simple graph with p vertices and q edges is complete
(has all possible edges) if and only if q=p(p-1)/2.
please prove it step by step. thanks

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