Question 1

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

Accepted Answer

$$\text { Vertices } = \{ 1,2,3,4 \} ; \text { Edges } = \{ \{ 1,2 \} , \{ 2,3 \} , \{ 3,4 \} , \{ 4,1 \} \} ; \left( \begin{array} { c c c c } 
0 & 1 & 0 & 0 \
0 & 0 & 1 & 0 \
0 & 0 & 0 & 1 \
1 & 0 & 0 & 0
\end{array} \right)$$

Question 2

In questions ﬁll in the blanks.1113:1156
-There are non-isomorphic simple graphs with 3 vertices.

Accepted Answer

4.

Question 3

In questions ﬁll in the blanks.1113:1156
-List all positive integers n such that $K _ { n } \text { has a Hamilton circuit but no Euler circuit. }$$\underline{\quad\quad}$

Accepted Answer

$$n \text { even } ( \neq 2 )$$

Question 4

In questions ﬁll in the blanks.1113:1156
-$K _ { m , n }$ has$\underline{\quad\quad}$ edges and$\underline{\quad\quad}$ vertices.

Accepted Answer

The answer of In questions ﬁll in the blanks.1113:1156
-\(K _...

Question 5

In questions either give an example or prove that there are none.
-A graph with 7 vertices that has a Hamilton circuit but no Euler circuit.

Accepted Answer

The answer of In questions either give an example or...

Question 6

In questions either give an example or prove that there are none.
-A simple graph with 6 vertices and 16 edges.

Accepted Answer

None. The largest nu

Question 7

If a simple graph G has v vertices and e edges, how many edges does G have?

Accepted Answer

The answer of If a simple graph G has v...

Question 8

Determine whether this graph is planar.   an oﬃce. Use graph theory ideas to prove that it
is impossible to plan a walk that passes through
each doorway exactly once, starting and ending at 
A.

Accepted Answer

The answer of Determine whether this graph is planar. ...

Question 9

In questions ﬁll in the blanks.1113:1156
-List all positive integers $$n \text { such that } C _ { n } \text { is bipartite }$$ .

Accepted Answer

The answer of In questions ﬁll in the blanks.1113:1156
-List all...

Question 10

In questions ﬁll in the blanks.1113:1156
-The adjacency matrix for $$Q _ { 4 } \text { has } \quad \text { entries. }$$

Accepted Answer

The answer of In questions ﬁll in the blanks.1113:1156
-The adjacency...

Question 11

In questions ﬁll in the blanks.1113:1156
-The incidence matrix for $Q _ { 5 }$ has $\underline{\quad\quad}$rows and$\underline{\quad\quad}$ columns.

Accepted Answer

The answer of In questions ﬁll in the blanks.1113:1156
-The incidence...

Question 12

Find the vertex-chromatic number, the edge-chromatic number, and the region-chromatic number for $W _ { 5 }$

Accepted Answer

vertex-chromatic number = 4; e

Question 13

In questions ﬁll in the blanks.1113:1156
-The adjacency matrix for $K _ { n }$ has $\underline{\quad\quad}$1 's and $\underline{\quad\quad}$0's.

Accepted Answer

The answer of In questions ﬁll in the blanks.1113:1156
-The adjacency...

Question 14

In questions ﬁll in the blanks.1113:1156
-There are non-isomorphic simple digraphs with 3 vertices and 2 edges.

Accepted Answer

The answer of In questions ﬁll in the blanks.1113:1156
-There are...

Question 15

Find the edge chromatic number for each of the graphs
-

Accepted Answer

The answer of Find the edge chromatic number for each...

Question 16

In questions ﬁll in the blanks.1113:1156
-$\text { List all positive integers } n \text { such that } W _ { n } \text { has an Euler circuit. }$$\underline{\quad\quad}$

Accepted Answer

The answer of In questions ﬁll in the blanks.1113:1156
-\(\text {...

Question 17

In questions ﬁll in the blanks.1113:1156
-$\text { Every Euler circuit for } K _ { 9 } \text { has length }$$\underline{\quad\quad}$

Accepted Answer

The answer of In questions ﬁll in the blanks.1113:1156
-\(\text {...

Question 18

In questions ﬁll in the blanks.1113:1156
-The length of the longest simple circuit in $$K _ { 5 } \text { is }$$ .

Accepted Answer

The answer of In questions ﬁll in the blanks.1113:1156
-The length...

Question 19

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

Accepted Answer

None. The sum of the

Question 20

Find the edge chromatic number for each of the graphs
-

Accepted Answer

3
In quest

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

In questions ﬁll in the blanks.1113:1156 -There are non-isomorphic simple graphs with 3 vertices.

In questions ﬁll in the blanks.1113:1156 -List all positive integers n such that $K _ { n } \text { has a Hamilton circuit but no Euler circuit. }$ $\underline{\quad\quad}$

In questions ﬁll in the blanks.1113:1156 - $K _ { m , n }$ has $\underline{\quad\quad}$ edges and $\underline{\quad\quad}$ vertices.

In questions either give an example or prove that there are none. -A graph with 7 vertices that has a Hamilton circuit but no Euler circuit.

In questions either give an example or prove that there are none. -A simple graph with 6 vertices and 16 edges.

If a simple graph G has v vertices and e edges, how many edges does G have?

Determine whether this graph is planar. an oﬃce. Use graph theory ideas to prove that it is impossible to plan a walk that passes through each doorway exactly once, starting and ending at A.

In questions ﬁll in the blanks.1113:1156 -List all positive integers $n \text { such that } C _ { n } \text { is bipartite }$ .

In questions ﬁll in the blanks.1113:1156 -The adjacency matrix for $Q _ { 4 } \text { has } \quad \text { entries. }$

In questions ﬁll in the blanks.1113:1156 -The incidence matrix for $Q _ { 5 }$ has $\underline{\quad\quad}$ rows and $\underline{\quad\quad}$ columns.

Find the vertex-chromatic number, the edge-chromatic number, and the region-chromatic number for $W _ { 5 }$

In questions ﬁll in the blanks.1113:1156 -The adjacency matrix for $K _ { n }$ has $\underline{\quad\quad}$ 1 's and $\underline{\quad\quad}$ 0's.

In questions ﬁll in the blanks.1113:1156 -There are non-isomorphic simple digraphs with 3 vertices and 2 edges.

Find the edge chromatic number for each of the graphs -

In questions ﬁll in the blanks.1113:1156 - $\text { List all positive integers } n \text { such that } W _ { n } \text { has an Euler circuit. }$ $\underline{\quad\quad}$

In questions ﬁll in the blanks.1113:1156 - $\text { Every Euler circuit for } K _ { 9 } \text { has length }$ $\underline{\quad\quad}$

In questions ﬁll in the blanks.1113:1156 -The length of the longest simple circuit in $K _ { 5 } \text { is }$ .

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

Find the edge chromatic number for each of the graphs -

The Foundations: Logic and Proofs

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

Algorithms

Number Theory and Cryptography

Induction and Recursion

Counting

Discrete Probability

Advanced Counting Techniques

Relations

Trees

Boolean Algebra

Modeling Computation

Filters

Exam 10: Graphs

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

In questions ﬁll in the blanks.1113:1156 -There are non-isomorphic simple graphs with 3 vertices.

In questions ﬁll in the blanks.1113:1156 -List all positive integers n such that Kn has a Hamilton circuit but no Euler circuit. K _ { n } \text { has a Hamilton circuit but no Euler circuit. }Kn​ has a Hamilton circuit but no Euler circuit. ‾\underline{\quad\quad}​

In questions ﬁll in the blanks.1113:1156 - Km,nK _ { m , n }Km,n​ has ‾\underline{\quad\quad}​ edges and ‾\underline{\quad\quad}​ vertices.

In questions either give an example or prove that there are none. -A graph with 7 vertices that has a Hamilton circuit but no Euler circuit.

In questions either give an example or prove that there are none. -A simple graph with 6 vertices and 16 edges.

If a simple graph G has v vertices and e edges, how many edges does G have?

Determine whether this graph is planar. an oﬃce. Use graph theory ideas to prove that it is impossible to plan a walk that passes through each doorway exactly once, starting and ending at A.

In questions ﬁll in the blanks.1113:1156 -List all positive integers n such that Cn is bipartite n \text { such that } C _ { n } \text { is bipartite }n such that Cn​ is bipartite .

In questions ﬁll in the blanks.1113:1156 -The adjacency matrix for Q4 has entries. Q _ { 4 } \text { has } \quad \text { entries. }Q4​ has entries.

In questions ﬁll in the blanks.1113:1156 -The incidence matrix for Q5Q _ { 5 }Q5​ has ‾\underline{\quad\quad}​ rows and ‾\underline{\quad\quad}​ columns.

Find the vertex-chromatic number, the edge-chromatic number, and the region-chromatic number for W5W _ { 5 }W5​

In questions ﬁll in the blanks.1113:1156 -The adjacency matrix for KnK _ { n }Kn​ has ‾\underline{\quad\quad}​ 1 's and ‾\underline{\quad\quad}​ 0's.

In questions ﬁll in the blanks.1113:1156 -There are non-isomorphic simple digraphs with 3 vertices and 2 edges.

Find the edge chromatic number for each of the graphs -

In questions ﬁll in the blanks.1113:1156 - Every Euler circuit for K9 has length \text { Every Euler circuit for } K _ { 9 } \text { has length } Every Euler circuit for K9​ has length ‾\underline{\quad\quad}​

In questions ﬁll in the blanks.1113:1156 -The length of the longest simple circuit in K5 is K _ { 5 } \text { is }K5​ is .

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

Find the edge chromatic number for each of the graphs -

The Foundations: Logic and Proofs

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

Algorithms

Number Theory and Cryptography

Induction and Recursion

Counting

Discrete Probability

Advanced Counting Techniques

Relations

Trees

Boolean Algebra

Modeling Computation

Filters

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

In questions ﬁll in the blanks.1113:1156 -List all positive integers n such that $K _ { n } \text { has a Hamilton circuit but no Euler circuit. }$ $\underline{\quad\quad}$

In questions ﬁll in the blanks.1113:1156 - $K _ { m , n }$ has $\underline{\quad\quad}$ edges and $\underline{\quad\quad}$ vertices.

In questions ﬁll in the blanks.1113:1156 -List all positive integers $n \text { such that } C _ { n } \text { is bipartite }$ .

In questions ﬁll in the blanks.1113:1156 -The adjacency matrix for $Q _ { 4 } \text { has } \quad \text { entries. }$

In questions ﬁll in the blanks.1113:1156 -The incidence matrix for $Q _ { 5 }$ has $\underline{\quad\quad}$ rows and $\underline{\quad\quad}$ columns.

Find the vertex-chromatic number, the edge-chromatic number, and the region-chromatic number for $W _ { 5 }$

In questions ﬁll in the blanks.1113:1156 -The adjacency matrix for $K _ { n }$ has $\underline{\quad\quad}$ 1 's and $\underline{\quad\quad}$ 0's.

In questions ﬁll in the blanks.1113:1156 - $\text { Every Euler circuit for } K _ { 9 } \text { has length }$ $\underline{\quad\quad}$

In questions ﬁll in the blanks.1113:1156 -The length of the longest simple circuit in $K _ { 5 } \text { is }$ .