Question 1

Draw the digraph with adjacency matrix $$\left( \begin{array} { l l l l } 
0 & 0 & 0 & 0 \
0 & 0 & 1 & 0 \
1 & 1 & 0 & 1 \
1 & 1 & 1 & 0
\end{array} \right)$$

Accepted Answer

The answer of Draw the digraph with adjacency matrix \[\left(...

Question 2

either give an example or prove that there are none.
-A graph with vertex-chromatic number equal to 6.

Accepted Answer

$K _ { 6 }$

Question 3

Solve the traveling salesman problem for the graph at the right by ﬁnding the total weight of all Hamilton circuits and determining a circuit with minimum total weight.

Accepted Answer

A-B-D-C-A (weight 19)

Question 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.
-$W _ { 5 }$

Accepted Answer

\[\begin{array} { l } 
\text { Vertices

Question 5

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

Accepted Answer

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

Question 6

ﬁll in the blanks.
-The incidence matrix for $W _ { n }$ has ____ rows and ____ columns.

Accepted Answer

The answer of ﬁll in the blanks.
-The incidence matrix for...

Question 7

Give a recurrence relation for en = number of edges of the graph $Q _ { n }$

Accepted Answer

Question 8

Find the vertex-chromatic number, the edge-chromatic number, and the region-chromatic number for C7.

Accepted Answer

vertex-chromatic num

Question 9

ﬁll in the blanks.
-If G is a connected graph with 12 regions and 20 edges, then G has ____ vertices.

Accepted Answer

The answer of ﬁll in the blanks.
-If G is a...

Question 10

either give an example or prove that there are none.
-A graph with 4 vertices that is not planar.

Accepted Answer

None. The

Question 11

Are these two graphs isomorphic?

Accepted Answer

The graphs are not isomorphic:

Question 12

Give a recurrence relation for en = the number of edges of the graph $W _ { n }$ circuit with minimum total weight.

Accepted Answer

Question 13

ﬁll in the blanks.
-The largest value of n for which $K _ { 6 , n }$ is planar is ____.

Accepted Answer

The answer of ﬁll in the blanks.
-The largest value of...

Question 14

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

Accepted Answer

m = n + 1

Question 15

ﬁll in the blanks.
-List all positive integers n such that $Q _ { n }$ has an Euler circuit ____.

Accepted Answer

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

Question 16

ﬁll in the blanks.
-The adjacency matrix for $K _ { n }$ has ____ 1's and ____ 0's.

Accepted Answer

The answer of ﬁll in the blanks.
-The adjacency matrix for...

Question 17

Use Dijkstra's Algorithm to ﬁnd the shortest path length between the vertices a and z in this weighted graph.

Accepted Answer

First iteration: distinguished vertices

Question 18

Find the strongly connected components of the graph.

Accepted Answer

The circuit a, f, b, c, f, b,

Question 19

In K3,3 let a and b be any two adjacent vertices. Find the number of paths between a and b of length 4.

Accepted Answer

Question 20

Find the edge chromatic number for each of the graphs
-

Accepted Answer

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

Exam 10: A: Graphs

Draw the digraph with adjacency matrix (0000001011011110)\left( \begin{array} { l l l l } 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 \\1 & 1 & 0 & 1 \\1 & 1 & 1 & 0\end{array} \right)​0011​0011​0101​0010​​

either give an example or prove that there are none. -A graph with vertex-chromatic number equal to 6.

Solve the traveling salesman problem for the graph at the right by ﬁnding the total weight of all Hamilton circuits and determining a circuit with minimum total weight.

for each graph give an ordered pair description (vertex set and edge set) and an adjacency matrix, and draw a picture of the graph. - W5W _ { 5 }W5​

ﬁll in the blanks. -The vertex-chromatic number for KnK _ { n }Kn​ = ____.

ﬁll in the blanks. -The incidence matrix for WnW _ { n }Wn​ has ____ rows and ____ columns.

Give a recurrence relation for en = number of edges of the graph QnQ _ { n }Qn​

Find the vertex-chromatic number, the edge-chromatic number, and the region-chromatic number for C7.

ﬁll in the blanks. -If G is a connected graph with 12 regions and 20 edges, then G has ____ vertices.

either give an example or prove that there are none. -A graph with 4 vertices that is not planar.

Are these two graphs isomorphic?

Give a recurrence relation for en = the number of edges of the graph WnW _ { n }Wn​ circuit with minimum total weight.

ﬁll in the blanks. -The largest value of n for which K6,nK _ { 6 , n }K6,n​ is planar is ____.

ﬁll in the blanks. -List all positive integers m and n such that Km,nK _ { m , n }Km,n​ has a Hamilton path but no Hamilton circuit.____

ﬁll in the blanks. -List all positive integers n such that QnQ _ { n }Qn​ has an Euler circuit ____.

ﬁll in the blanks. -The adjacency matrix for KnK _ { n }Kn​ has ____ 1's and ____ 0's.

Use Dijkstra's Algorithm to ﬁnd the shortest path length between the vertices a and z in this weighted graph.

Find the strongly connected components of the graph.

In K3,3 let a and b be any two adjacent vertices. Find the number of paths between a and b of length 4.

Find the edge chromatic number for each of the graphs -

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

Draw the digraph with adjacency matrix $\left( \begin{array} { l l l l } 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 \\1 & 1 & 0 & 1 \\1 & 1 & 1 & 0\end{array} \right)$

for each graph give an ordered pair description (vertex set and edge set) and an adjacency matrix, and draw a picture of the graph. - $W _ { 5 }$

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

ﬁll in the blanks. -The incidence matrix for $W _ { n }$ has rows and columns.

Give a recurrence relation for e_n = number of edges of the graph $Q _ { n }$

Find the vertex-chromatic number, the edge-chromatic number, and the region-chromatic number for C₇.

Give a recurrence relation for e_n = the number of edges of the graph $W _ { n }$ circuit with minimum total weight.

ﬁll in the blanks. -The largest value of n for which $K _ { 6 , n }$ is planar is ____.

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

ﬁll in the blanks. -List all positive integers n such that $Q _ { n }$ has an Euler circuit ____.

ﬁll in the blanks. -The adjacency matrix for $K _ { n }$ has 1's and 0's.

In K_3,3 let a and b be any two adjacent vertices. Find the number of paths between a and b of length 4.