Question 1

Draw a directed graph with the following adjacency matrix:

Accepted Answer

The answer of Draw a directed graph with the following...

Question 2

$\text { Consider the following weighted graph: }$

(a) Use Kruskal's algorithm to ﬁnd a minimum spanning tree for the graph, and indicate the
order in which edges are added to form the tree.
(b) Use Prim's algorithm starting with vertex a to ﬁnd a minimum spanning tree for the
graph, and indicate the order in which edges are added to form the tree.
(c) Use Dijkstra's algorithm to ﬁnd the shortest path from a to

Accepted Answer

a. Order of adding the edges: $\{ c , d \} , \{ d , f \} , \{ b , c \} , \{ b , e \} , \{ f , g \} , \{ a , e \} , \{ a , g \}$ or $\{ c , d \} , \{ d , f \} , \{ b , c \} , \{ f , g \} , \{ b , e \} , \{ a , e \} , \{ a , g \}$
b. Order of adding the edges: $\{ a , e \} , \{ b , e \} , \{ b , c \} , \{ c , d \} , \{ d , f \} , \{ f , g \}$
c. Dijkstra's algorithm shows that the shortest path from $a$ to $d$ has length 15 . Because $D ( d ) = c , D ( c ) = e$, and $D ( e ) = a$, we can work backward to find that the path is $a , e , c , d$.

Question 3

Determine whether each of the following graphs has an Euler circuit. If it does have an Euler circuit, ﬁnd such a circuit. If it does not have an Euler circuit, explain why you can be 100% sure that it does not.

Accepted Answer

$G _ { 1 }$ (the left-hand graph) does not have an Euler circuit because it has a vertex of odd degree. In fact both vertices $g$ and $e$ have odd degree.
$G _ { 2 }$ (the right-hand graph) does not have an Euler circuit because it has a vertex of odd degree. In fact both vertices $b$ and $l$ have odd degree.

Question 4

Find the following matrix product: $$\left[ \begin{array} { l l } 
2 & 0 \
0 & 1 \
3 & 2
\end{array} \right] \left[ \begin{array} { l l l } 
1 & 3 & 0 \
2 & 4 & 2
\end{array} \right]$$

Accepted Answer

None

Question 5

Determine whether each of the following graphs has a Hamiltonian circuit. If it does have an Hamiltonian circuit, ﬁnd such a circuit. If it does not have an Hamiltonian circuit, explain why you can be 100% sure that it does not.

Accepted Answer

Suppose @#LAT-DLM& (the left-hand graph) has a Ham

Question 6

a. Prove that having n vertices, where n is a positive integer, is an invariant for graph
isomorphism.
b. Prove that having a vertex of degree 3 is an invariant for graph isomorphism.

Accepted Answer

a. Proof:
Suppose @#LAT-DLM& and @#LAT-DLM& are isomorphic g

Question 7

A certain connected graph has 68 vertices and 72 edges. Does it have a circuit? Explain.

Accepted Answer

Yes. The graph does has a circ

Question 8

Consider the adjacency matrix for a graph that is shown below. Answer the following questions by examining the matrix and its powers only, not by drawing the graph. Show your work in a way that makes your reasoning clear.   
 (a) How many walks of length 2 are there from $v _ { 1 }$ to $v _ { 2 }$ ?
(b) How many walks of length 2 are there from $v _ { 1 }$ to $v _ { 3 }$ ?
(c) How many walks of length 2 are there from $v _ { 2 }$ to $v _ { 2 }$ ?

Accepted Answer

Let \[A = \left[ \begin{array} { l l l l

Question 9

Either draw a graph with the given speciﬁcation or explain why no such graph exists.
(a) full binary tree with 16 vertices of which 6 are internal vertices
(b) binary tree, height 3, 9 vertices
(c) binary tree, height 4, 18 terminal vertices

Accepted Answer

a. No such graph exists because a full b

Question 10

A certain graph has 19 vertices, 16 edges, and no circuits. Is it connected? Explain.

Accepted Answer

No. The graph is not connected

Question 11

For each of (a)-(c) below, either draw a graph with the speciﬁed properties or else explain why no such graph exists.
(a) Graph with six vertices of degrees 1, 1, 2, 2, 2, and 3.
(b) Graph with four vertices of degrees 1, 2, 2, and 5.
(c) Simple graph with four vertices of degrees 1, 1, 1, and 5.

Accepted Answer

a. Given any graph, the total degree of

Question 12

Determine whether any two of $G _ { 1 } , G _ { 2 }$, and $G _ { 3 }$ are isomorphic. If they are, give vertex and edge functions that define the isomorphism. If they are not, give an isomorphic invariant that they do not share.

Accepted Answer

@#LAT-DLM& is not isomorphic to @#LAT-DLM& because @#LAT-DLM& has one

Question 13

If a graph has vertices of degrees 1, 1, 2, 3, and 3, how many edges does it have? Why?

Accepted Answer

In general, the number of edge

Question 14

Determine whether any two of the simple graphs $G _ { 1 } , G _ { 2 }$, and $G _ { 3 }$ are isomorphic. If they are, give a vertex function that defines the isomorphism. If they are not, give an isomorphic invariant that they do not share.

Accepted Answer

@#LAT-DLM& is not isomorphic to @#LAT-DLM& because @#LAT-DLM& has a ci

Draw a directed graph with the following adjacency matrix:

Determine whether each of the following graphs has an Euler circuit. If it does have an Euler circuit, ﬁnd such a circuit. If it does not have an Euler circuit, explain why you can be 100% sure that it does not.

Find the following matrix product: $\left[ \begin{array} { l l } 2 & 0 \\0 & 1 \\3 & 2\end{array} \right] \left[ \begin{array} { l l l } 1 & 3 & 0 \\2 & 4 & 2\end{array} \right]$

Determine whether each of the following graphs has a Hamiltonian circuit. If it does have an Hamiltonian circuit, ﬁnd such a circuit. If it does not have an Hamiltonian circuit, explain why you can be 100% sure that it does not.

a. Prove that having n vertices, where n is a positive integer, is an invariant for graph isomorphism. b. Prove that having a vertex of degree 3 is an invariant for graph isomorphism.

A certain connected graph has 68 vertices and 72 edges. Does it have a circuit? Explain.

Either draw a graph with the given speciﬁcation or explain why no such graph exists. (a) full binary tree with 16 vertices of which 6 are internal vertices (b) binary tree, height 3, 9 vertices (c) binary tree, height 4, 18 terminal vertices

A certain graph has 19 vertices, 16 edges, and no circuits. Is it connected? Explain.

For each of (a)-(c) below, either draw a graph with the speciﬁed properties or else explain why no such graph exists. (a) Graph with six vertices of degrees 1, 1, 2, 2, 2, and 3. (b) Graph with four vertices of degrees 1, 2, 2, and 5. (c) Simple graph with four vertices of degrees 1, 1, 1, and 5.

If a graph has vertices of degrees 1, 1, 2, 3, and 3, how many edges does it have? Why?

Speaking Mathematically

The Logic of Compound Statements

The Logic of Quantified Statements

Elementary Number Theory and Methods of Proof

Sequences, Mathematical Induction, and Recursion

Set Theory

Functions

Relations

Counting and Probability

Analyzing Algorithm Efficiency

Regular Expressions and Finite State Automata

Filters

Exam 10: Graphs and Trees

Draw a directed graph with the following adjacency matrix:

Determine whether each of the following graphs has an Euler circuit. If it does have an Euler circuit, ﬁnd such a circuit. If it does not have an Euler circuit, explain why you can be 100% sure that it does not.

Find the following matrix product: [200132][130242]\left[ \begin{array} { l l } 2 & 0 \\0 & 1 \\3 & 2\end{array} \right] \left[ \begin{array} { l l l } 1 & 3 & 0 \\2 & 4 & 2\end{array} \right]​203​012​​[12​34​02​]

Determine whether each of the following graphs has a Hamiltonian circuit. If it does have an Hamiltonian circuit, ﬁnd such a circuit. If it does not have an Hamiltonian circuit, explain why you can be 100% sure that it does not.

a. Prove that having n vertices, where n is a positive integer, is an invariant for graph isomorphism. b. Prove that having a vertex of degree 3 is an invariant for graph isomorphism.

A certain connected graph has 68 vertices and 72 edges. Does it have a circuit? Explain.

Either draw a graph with the given speciﬁcation or explain why no such graph exists. (a) full binary tree with 16 vertices of which 6 are internal vertices (b) binary tree, height 3, 9 vertices (c) binary tree, height 4, 18 terminal vertices

A certain graph has 19 vertices, 16 edges, and no circuits. Is it connected? Explain.

For each of (a)-(c) below, either draw a graph with the speciﬁed properties or else explain why no such graph exists. (a) Graph with six vertices of degrees 1, 1, 2, 2, 2, and 3. (b) Graph with four vertices of degrees 1, 2, 2, and 5. (c) Simple graph with four vertices of degrees 1, 1, 1, and 5.

Determine whether any two of G1,G2G _ { 1 } , G _ { 2 }G1​,G2​ , and G3G _ { 3 }G3​ are isomorphic. If they are, give vertex and edge functions that define the isomorphism. If they are not, give an isomorphic invariant that they do not share.

If a graph has vertices of degrees 1, 1, 2, 3, and 3, how many edges does it have? Why?

Determine whether any two of the simple graphs G1,G2G _ { 1 } , G _ { 2 }G1​,G2​ , and G3G _ { 3 }G3​ are isomorphic. If they are, give a vertex function that defines the isomorphism. If they are not, give an isomorphic invariant that they do not share.

Speaking Mathematically

The Logic of Compound Statements

The Logic of Quantified Statements

Elementary Number Theory and Methods of Proof

Sequences, Mathematical Induction, and Recursion

Set Theory

Functions

Relations

Counting and Probability

Analyzing Algorithm Efficiency

Regular Expressions and Finite State Automata

Filters

Find the following matrix product: $\left[ \begin{array} { l l } 2 & 0 \\0 & 1 \\3 & 2\end{array} \right] \left[ \begin{array} { l l l } 1 & 3 & 0 \\2 & 4 & 2\end{array} \right]$