Question 1

ﬁll in the blanks.
-There are ____ non-isomorphic simple undirected graphs with 5 vertices and 3 edges.

Accepted Answer

The answer of ﬁll in the blanks.
-There are ____ non-isomorphic...

Question 2

ﬁll in the blanks.
-List all positive integers m and n such that $K _ { m , n }$ has a Hamilton circuit.____

Accepted Answer

The answer of ﬁll in the blanks.
-List all positive integers...

Question 3

ﬁll in the blanks.
-The vertex-chromatic number for $C _ { 15 }$ = ____.

Accepted Answer

The answer of ﬁll in the blanks.
-The vertex-chromatic number for...

Question 4

either give an example or prove that there are none.
-A simple graph with degrees 1, 2, 2, 3.

Accepted Answer

The answer of either give an example or prove that...

Question 5

ﬁll in the blanks.
-$K _ { m , n }$ has ____ edges and ____ vertices.

Accepted Answer

The answer of ﬁll in the blanks.
-\(K _ { m...

Question 6

ﬁll in the blanks.
-$K _ { n }$ has ____ edges and ____ vertices.

Accepted Answer

The answer of ﬁll in the blanks.
-\(K _ { n...

Question 7

Are these two graphs isomorphic?

Accepted Answer

The graphs are isomo

Question 8

either give an example or prove that there are none.
-A simple digraph with indegrees: 0, 1, 2, 2, 3, 4 and outdegrees: 1, 1, 2, 2, 3, 4.

Accepted Answer

None. The sum of the

Question 9

for each graph give an ordered pair description (vertex set and edge set) and an adjacency matrix, and draw a picture of the graph.
-$Q _ { 3 }$

Accepted Answer

Vertices @#LAT-DLM&
Edges @#LAT-DLM&
\[\left( \begin{array}

Question 10

Determine whether the graph is strongly connected, and if not, whether it is weakly connected.

Accepted Answer

The circuits a, e, c

Question 11

ﬁll in the blanks.
-List all positive integers n such that $K _ { n }$ has a Hamilton circuit but no Euler circuit.____

Accepted Answer

The answer of ﬁll in the blanks.
-List all positive integers...

Question 12

ﬁll in the blanks.
-The Euler formula for planar connected graphs states that ____.

Accepted Answer

The answer of ﬁll in the blanks.
-The Euler formula for...

Question 13

ﬁll in the blanks.
-The edge-chromatic number for $K _ { 2.5 }$ = ____.

Accepted Answer

The answer of ﬁll in the blanks.
-The edge-chromatic number for...

Question 14

In K3,3 let a and b be any two adjacent vertices. Find the number of paths between a and b of length 5.

Accepted Answer

Question 15

either give an example or prove that there are none.
-A simple graph with 6 vertices, whose degrees are 2, 2, 2, 3, 4, 4.

Accepted Answer

None. It is not poss

Question 16

either give an example or prove that there are none.
-A bipartite graph with an odd number of vertices that has a Hamilton circuit.

Accepted Answer

None. Any bipartite

Question 17

Explain how graphs can be used to model the spread of a contagious disease. Should the edges be directed or undirected? Should multiple edges be allowed? Should loops be allowed?

Accepted Answer

Use a directed graph, with the vertices

Question 18

either give an example or prove that there are none.
-A planar graph with 10 vertices.

Accepted Answer

The answer of either give an example or prove that...

Question 19

for each graph give an ordered pair description (vertex set and edge set) and an adjacency matrix, and draw a picture of the graph.
-$C _ { 4 }$

Accepted Answer

\[\text { Vertices } = \{ 1,2,3,4 \} ; \

Question 20

either give an example or prove that there are none.
-A graph with 6 vertices that has an Euler circuit but no Hamilton circuit.

Accepted Answer

The answer of either give an example or prove that...

Exam 10: A: Graphs

ﬁll in the blanks. -There are ____ non-isomorphic simple undirected graphs with 5 vertices and 3 edges.

ﬁll in the blanks. -List all positive integers m and n such that Km,nK _ { m , n }Km,n​ has a Hamilton circuit.____

ﬁll in the blanks. -The vertex-chromatic number for C15C _ { 15 }C15​ = ____.

either give an example or prove that there are none. -A simple graph with degrees 1, 2, 2, 3.

ﬁll in the blanks. - Km,nK _ { m , n }Km,n​ has ____ edges and ____ vertices.

ﬁll in the blanks. - KnK _ { n }Kn​ has ____ edges and ____ vertices.

Are these two graphs isomorphic?

either give an example or prove that there are none. -A simple digraph with indegrees: 0, 1, 2, 2, 3, 4 and outdegrees: 1, 1, 2, 2, 3, 4.

for each graph give an ordered pair description (vertex set and edge set) and an adjacency matrix, and draw a picture of the graph. - Q3Q _ { 3 }Q3​

Determine whether the graph is strongly connected, and if not, whether it is weakly connected.

ﬁll in the blanks. -List all positive integers n such that KnK _ { n }Kn​ has a Hamilton circuit but no Euler circuit.____

ﬁll in the blanks. -The Euler formula for planar connected graphs states that ____.

ﬁll in the blanks. -The edge-chromatic number for K2.5K _ { 2.5 }K2.5​ = ____.

In K3,3 let a and b be any two adjacent vertices. Find the number of paths between a and b of length 5.

either give an example or prove that there are none. -A simple graph with 6 vertices, whose degrees are 2, 2, 2, 3, 4, 4.

either give an example or prove that there are none. -A bipartite graph with an odd number of vertices that has a Hamilton circuit.

Explain how graphs can be used to model the spread of a contagious disease. Should the edges be directed or undirected? Should multiple edges be allowed? Should loops be allowed?

either give an example or prove that there are none. -A planar graph with 10 vertices.

for each graph give an ordered pair description (vertex set and edge set) and an adjacency matrix, and draw a picture of the graph. - C4C _ { 4 }C4​

either give an example or prove that there are none. -A graph with 6 vertices that has an Euler circuit but no Hamilton circuit.

The Foundations: Logic and Proofs

A: the Foundations: Logic and Proofs

Basic Structures: Sets, Functions, Sequences, Sums, Matrices

A: Basic Structures: Sets, Functions, Sequences, Sums, Matrices

Algorithms

A: Algorithms

Number Theory and Cryptography

A: Number Theory and Cryptography

Induction and Recursion

A: Induction and Recursion

Counting

A: Counting

Discrete Probability

A: Discrete Probability

Advanced Counting Techniques

A: Advanced Counting Techniques

Relations

A: Relations

Graphs

Trees

A: Trees

Boolean Algebra

A: Boolean Algebra

Modeling Computation

A: Modeling Computation

Mathematics Problem Set: Set Theory, Number Theory, Combinatorics, and Boolean Algebra

Filters

ﬁll in the blanks. -List all positive integers m and n such that $K _ { m , n }$ has a Hamilton circuit.____

ﬁll in the blanks. -The vertex-chromatic number for $C _ { 15 }$ = ____.

ﬁll in the blanks. - $K _ { m , n }$ has edges and vertices.

ﬁll in the blanks. - $K _ { n }$ has edges and vertices.

for each graph give an ordered pair description (vertex set and edge set) and an adjacency matrix, and draw a picture of the graph. - $Q _ { 3 }$

ﬁll in the blanks. -List all positive integers n such that $K _ { n }$ has a Hamilton circuit but no Euler circuit.____

ﬁll in the blanks. -The edge-chromatic number for $K _ { 2.5 }$ = ____.

In K_3,3 let a and b be any two adjacent vertices. Find the number of paths between a and b of length 5.

for each graph give an ordered pair description (vertex set and edge set) and an adjacency matrix, and draw a picture of the graph. - $C _ { 4 }$