Graphs and Graph Models Further Reading: sec. 10.1 of Rosen 1 A - - PowerPoint PPT Presentation

1

Graphs and Graph Models

Further Reading: sec. 10.1 of Rosen

2

A graph is a mathematical structure consisting of:

• Mathematical objects we call vertices.
• Mathematical objects we call edges, each of which

the graph associates with a pair of vertices that are called the endpoints of the edge.

3

A graph is a mathematical structure consisting of:

• Mathematical objects we call vertices.
• Mathematical objects we call edges, each of which

the graph associates with a pair of vertices that are called the endpoints of the edge.

4

A graph is a mathematical structure consisting of:

• Mathematical objects we call vertices.
• Mathematical objects we call edges, each of which

the graph associates with two (not necessarily distinct) vertices that are called the endpoints of the edge.

5

A graph is a mathematical structure consisting of:

• Mathematical objects we call vertices.
• Mathematical objects we call edges, each of which

the graph associates with two (not necessarily distinct) vertices that are called the endpoints of the edge. We say an edge is directed if one endpoint is designated as the initial endpoint and one as the terminal endpoint.

• In an undirected graph, no edge is directed.
• In a directed graph or digraph, every edge is directed.

6

A graph is a mathematical structure consisting of:

• Mathematical objects we call vertices.
• Mathematical objects we call edges, each of which

the graph associates with two (not necessarily distinct) vertices that are called the endpoints of the edge. We say an edge is directed if one endpoint is designated as the initial endpoint and one as the terminal endpoint.

• In an undirected graph, no edge is directed.
• In a directed graph or digraph, every edge is directed.

7

A graph is a mathematical structure consisting of:

• Mathematical objects we call vertices.
• Mathematical objects we call edges, each of which

the graph associates with two (not necessarily distinct) vertices that are called the endpoints of the edge. We say an edge is directed if one endpoint is designated as the initial endpoint and one as the terminal endpoint.

• In an undirected graph, no edge is directed.
• In a directed graph or digraph, every edge is directed.

8

A graph is a mathematical structure consisting of:

• Mathematical objects we call vertices.
• Mathematical objects we call edges, each of which

the graph associates with two (not necessarily distinct) vertices that are called the endpoints of the edge. We say an edge is directed if one endpoint is designated as the initial endpoint and one as the terminal endpoint.

• In an undirected graph, no edge is directed.
• In a directed graph or digraph, every edge is directed.

Graphs are often represented by drawings such as:

Drawing of an undirected graph that has 6 vertices

U, V, W, X, Y, Z

and has 14 edges

a, b, ..., n. Notation: means

"Vertices a and b are the endpoints of edge y".

9

A graph is a mathematical structure consisting of:

• Mathematical objects we call vertices.
• Mathematical objects we call edges, each of which

the graph associates with two (not necessarily distinct) vertices that are called the endpoints of the edge. We say an edge is directed if one endpoint is designated as the initial endpoint and one as the terminal endpoint.

• In an undirected graph, no edge is directed.
• In a directed graph or digraph, every edge is directed.

Graphs are often represented by drawings such as:

Drawing of an undirected graph that has 6 vertices

U, V, W, X, Y, Z

and has 14 edges

a, b, ..., n. Notation: means

"Vertices a and b are the endpoints of edge y".

10

A graph is a mathematical structure consisting of:

• Mathematical objects we call vertices.
• Mathematical objects we call edges, each of which

the graph associates with two (not necessarily distinct) vertices that are called the endpoints of the edge. We say an edge is directed if one endpoint is designated as the initial endpoint and one as the terminal endpoint.

• In an undirected graph, no edge is directed.
• In a directed graph or digraph, every edge is directed.

Graphs are often represented by drawings such as:

Drawing of an undirected graph that has 6 vertices

U, V, W, X, Y, Z

and has 14 edges

a, b, ..., n. Notation: means

"Vertices a and b are the endpoints of edge y".

11

A graph is a mathematical structure consisting of:

• Mathematical objects we call vertices.
• Mathematical objects we call edges, each of which

the graph associates with two (not necessarily distinct) vertices that are called the endpoints of the edge. We say an edge is directed if one endpoint is designated as the initial endpoint and one as the terminal endpoint.

• In an undirected graph, no edge is directed.
• In a directed graph or digraph, every edge is directed.

Graphs are often represented by drawings such as:

Drawing of an undirected graph that has 6 vertices

U, V, W, X, Y, Z

and has 14 edges

a, b, ..., n. Notation: means

"Vertices a and b are the endpoints of edge y".

12

A graph is a mathematical structure consisting of:

• Mathematical objects we call vertices.
• Mathematical objects we call edges, each of which

the graph associates with two (not necessarily distinct) vertices that are called the endpoints of the edge. We say an edge is directed if one endpoint is designated as the initial endpoint and one as the terminal endpoint.

• In an undirected graph, no edge is directed.
• In a directed graph or digraph, every edge is directed.

Graphs are often represented by drawings such as:

Drawing of an undirected graph that has 6 vertices

U, V, W, X, Y, Z

and has 14 edges

a, b, ..., n. Notation: means

"Vertices a and b are the endpoints of edge y".

13

A graph is a mathematical structure consisting of:

• Mathematical objects we call vertices.
• Mathematical objects we call edges, each of which

the graph associates with a pair of vertices that are called the endpoints of the edge. We say an edge is directed if one endpoint is designated as the initial endpoint and one as the terminal endpoint.

• In an undirected graph, no edge is directed.
• In a directed graph or digraph, every edge is directed.

Graphs are often represented by drawings such as:

Drawing of an undirected graph that has 6 vertices

U, V, W, X, Y, Z

and has 14 edges

a, b, ..., n. Notation: means

"Vertices a and b are the endpoints of edge y".

14

Drawing of an undirected graph that has 6 vertices

U, V, W, X, Y, Z

and has 14 edges

a, b, ..., n. Notation: means

"Vertices a and b are the endpoints of edge y".

15

Drawing of an undirected graph that has 6 vertices U, V, W, X, Y, Z and has 14 edges a, b, ..., n. Notation: means "Vertices a and b are the endpoints of edge y".

16

Drawing of an undirected graph that has 6 vertices U, V, W, X, Y, Z and has 14 edges a, b, ..., n. Notation: means "Vertices a and b are the endpoints of edge y".

Here is a drawing of a directed graph or digraph:

Drawing of a digraph that has 6 vertices U, V, W, X, Y, Z and has 14 edges a, b, ..., n. Notation: means "Vertex a is the initial endpoint and vertex b is the terminal endpoint

• f edge y".

17

Drawing of an undirected graph that has 6 vertices U, V, W, X, Y, Z and has 14 edges a, b, ..., n. Notation: means "Vertices a and b are the endpoints of edge y".

Here is a drawing of a directed graph or digraph:

Drawing of a digraph that has 6 vertices U, V, W, X, Y, Z and has 14 edges a, b, ..., n. Notation: means "Vertex a is the initial endpoint and vertex b is the terminal endpoint

• f edge y".

18

Drawing of an undirected graph that has 6 vertices U, V, W, X, Y, Z and has 14 edges a, b, ..., n. Notation: means "Vertices a and b are the endpoints of edge y".

Here is a drawing of a directed graph or digraph:

Drawing of a digraph that has 6 vertices U, V, W, X, Y, Z and has 14 edges a, b, ..., n. Notation: means "Vertex a is the initial endpoint and vertex b is the terminal endpoint

• f edge y".

19

Note: A graph drawing is not a graph: It's a representation

• f the mathematical object that is the graph itself.
• The relationship of a graph drawing to the graph it

depicts is akin to the relationship of a subway map to the subway system it depicts, and also akin to the relationship of a photo of a person to the actual person shown by the photo.

• It's very common to refer to a photo of a person as a

person: For example, someone might show a photo of his son and say "This is my son."

• Similarly, it's very common to refer to a graph drawing as

a graph: For example, you can point to a graph drawing and say "This is the graph described on page 4."

• This is fine provided we are aware that the drawing

is not really the graph.

• Just as different photos of the same person may look quite

different, different drawings of the same graph may look quite different!

20

Note: A graph drawing is not a graph: It's a representation

• f the mathematical object that is the graph itself.
• The relationship of a graph drawing to the graph it

depicts is akin to the relationship of a subway map to the subway system it depicts, and also akin to the relationship of a photo of a person to the actual person shown by the photo.

• It's very common to refer to a photo of a person as a

person: For example, someone might show a photo of his son and say "This is my son."

• Similarly, it's very common to refer to a graph drawing as

a graph: For example, you can point to a graph drawing and say "This is the graph described on page 4."

• This is fine provided we are aware that the drawing

is not really the graph.

• Just as different photos of the same person may look quite

different, different drawings of the same graph may look quite different!

21

Note: A graph drawing is not a graph: It's a representation

• f the mathematical object that is the graph itself.
• The relationship of a graph drawing to the graph it

depicts is akin to the relationship of a subway map to the subway system it depicts, and also akin to the relationship of a photo of a person to the actual person shown by the photo.

• It's very common to refer to a photo of a person as a

person: For example, someone might show a photo of his son and say "This is my son."

• Similarly, it's very common to refer to a graph drawing as

a graph: For example, you can point to a graph drawing and say "This is the graph described on page 4."

• This is fine provided we are aware that the drawing

is not really the graph.

• Just as different photos of the same person may look quite

different, different drawings of the same graph may look quite different!

22

Note: A graph drawing is not a graph: It's a representation

• f the mathematical object that is the graph itself.
• The relationship of a graph drawing to the graph it

depicts is akin to the relationship of a subway map to the subway system it depicts, and also akin to the relationship of a photo of a person to the actual person shown by the photo.

• It's very common to refer to a photo of a person as a

person: For example, someone might show a photo of his son and say "This is my son."

• Similarly, it's very common to refer to a graph drawing as

a graph: For example, you can point to a graph drawing and say "This is the graph described on page 4."

• This is fine provided we are aware that the drawing

is not really the graph.

• Just as different photos of the same person may look quite

different, different drawings of the same graph may look quite different!

23

Note: A graph drawing is not a graph: It's a representation

• f the mathematical object that is the graph itself.
• The relationship of a graph drawing to the graph it

depicts is akin to the relationship of a subway map to the subway system it depicts, and also akin to the relationship of a photo of a person to the actual person shown by the photo.

• It's very common to refer to a photo of a person as a

person: For example, someone might show a photo of his son and say "This is my son."

• Similarly, it's very common to refer to a graph drawing as

a graph: For example, you can point to a graph drawing and say "This is the graph described on page 4."

• This is fine provided we are aware that the drawing

is not really the graph.

• Just as different photos of the same person may look quite

different, different drawings of the same graph may look quite different!

24

Note: A graph drawing is not a graph: It's a representation

• f the mathematical object that is the graph itself.
• The relationship of a graph drawing to the graph it

depicts is akin to the relationship of a subway map to the subway system it depicts, and also akin to the relationship of a photo of a person to the actual person shown by the photo.

• It's very common to refer to a photo of a person as a

person: For example, someone might show a photo of his son and say "This is my son."

• Similarly, it's very common to refer to a graph drawing as

a graph: For example, you can point to a graph drawing and say "This is the graph described on page 4."

• This is fine provided we are aware that the drawing

is not really the graph.

• Just as different photos of the same person may look quite

different, different drawings of the same graph may look quite different!

25

Note: A graph drawing is not a graph: It's a representation

• f the mathematical object that is the graph itself.
• The relationship of a graph drawing to the graph it

depicts is akin to the relationship of a subway map to the subway system it depicts, and also akin to the relationship of a photo of a person to the actual person shown by the photo.

• It's very common to refer to a photo of a person as a

person: For example, someone might show a photo of his son and say "This is my son."

• Similarly, it's very common to refer to a graph drawing as

a graph: For example, you can point to a graph drawing and say "This is the graph described on page 4."

• This is fine provided we are aware that the drawing

is not really the graph.

• Just as different photos of the same person may look quite

different, different drawings of the same graph may look quite different, as the following examples illustrate!

26

The two graph drawings on the right are drawings of the same undirected graph, because:

• 1. The depicted graphs have

exactly the same vertices, namely U, V, W, X, Y, Z.

• 2. The depicted graphs have

exactly the same edges, namely a, b, ..., n.

• 3. Each of the 14 edges has

the same pair of endpoints in both of the depicted graphs.

27

The two graph drawings on the right are drawings of the same undirected graph, because:

• 1. The depicted graphs have

exactly the same vertices, namely U, V, W, X, Y, Z.

• 2. The depicted graphs have

exactly the same edges, namely a, b, ..., n.

• 3. Each of the 14 edges has

the same pair of endpoints in both of the depicted graphs.

28

The two graph drawings on the right are drawings of the same undirected graph, because:

• 1. The depicted graphs have

exactly the same vertices, namely U, V, W, X, Y, Z.

• 2. The depicted graphs have

exactly the same edges, namely a, b, ..., n.

• 3. Each of the 14 edges has

the same pair of endpoints in both of the depicted graphs.

29

The two graph drawings on the right are drawings of the same undirected graph, because:

• 1. The depicted graphs have

exactly the same vertices, namely U, V, W, X, Y, Z.

• 2. The depicted graphs have

exactly the same edges, namely a, b, ..., n.

• 3. Each of the 14 edges has

the same two endpoints in both of the depicted graphs.

30

Similarly, the two directed graph drawings on the right are drawings of the same directed graph, because:

• 1. The depicted digraphs have

exactly the same vertices, namely U, V, W, X, Y, Z.

• 2. The depicted digraphs have

exactly the same edges, namely a, b, ..., n.

• 3. Each of the 14 edges has

the same initial endpoint and the same terminal endpoint in both of the depicted digraphs.

31

Similarly, the two directed graph drawings on the right are drawings of the same directed graph, because:

• 1. The depicted digraphs have

exactly the same vertices, namely U, V, W, X, Y, Z.

• 2. The depicted digraphs have

exactly the same edges, namely a, b, ..., n.

• 3. Each of the 14 edges has

the same initial endpoint and the same terminal endpoint in both of the depicted digraphs.

32

Similarly, the two directed graph drawings on the right are drawings of the same directed graph, because:

• 1. The depicted digraphs have

exactly the same vertices, namely U, V, W, X, Y, Z.

• 2. The depicted digraphs have

exactly the same edges, namely a, b, ..., n.

• 3. Each of the 14 edges has

the same initial endpoint and the same terminal endpoint in both of the depicted digraphs.

33

Similarly, the two directed graph drawings on the right are drawings of the same directed graph, because:

• 1. The depicted digraphs have

exactly the same vertices, namely U, V, W, X, Y, Z.

• 2. The depicted digraphs have

exactly the same edges, namely a, b, ..., n.

• 3. Each of the 14 edges has

the same initial endpoint and the same terminal endpoint in both of the depicted digraphs.

34

Remark on Vertex and Edge Labels in Our Graph Drawings

• We have used graph drawings (such as

the drawing on the right) in which each vertex and each edge has a label that identifies it.

• But if we have no interest in the

identity of a vertex or edge, then we can omit its label.

35

Remark on Vertex and Edge Labels in Our Graph Drawings

• We have used graph drawings (such as

the drawing on the right) in which each vertex and each edge has a label that identifies it.

• But if we have no interest in the

identity of a vertex or edge, then we can omit its label. Example

36

Remark on Vertex and Edge Labels in Our Graph Drawings

• We have used graph drawings (such as

the drawing on the right) in which each vertex and each edge has a label that identifies it.

• But if we have no interest in the

identity of a vertex or edge, then we can omit its label. Example Suppose an easy exam question asks:

• How many edges of the digraph that is drawn below have

vertex U as their initial endpoint?"

37

Remark on Vertex and Edge Labels in Our Graph Drawings

• We have used graph drawings (such as

the drawing on the right) in which each vertex and each edge has a label that identifies it.

• But if we have no interest in the

identity of a vertex or edge, then we can omit its label. Example Suppose an easy exam question asks:

• How many edges of the digraph that is drawn below have

vertex U as their initial endpoint?" Then no edge in the drawing would need to be labeled, and U is the only vertex of the drawing that would need a label--so the drawing might be as shown on here:

38

Remark on Vertex and Edge Labels in Our Graph Drawings

• We have used graph drawings (such as

the drawing on the right) in which each vertex and each edge has a label that identifies it.

• But if we have no interest in the

identity of a vertex or edge, then we can omit its label. Example Suppose an easy exam question asks:

• How many edges of the digraph that is drawn below have

vertex U as their initial endpoint?" Then no edge in the drawing would need to be labeled, and U is the only vertex of the drawing that would need a label--so the drawing might be as shown on here: If the question merely asked "How many edges does this digraph have?", then no vertex or edge of the drawing would need a label!

39

Graphs have numerous applications in many fields.

• Partly as a result of this, graph terminology is

not well standardized!

• Two Examples: What we call a vertex is sometimes

called a node or a point.

called a node or a point. What we call an edge is sometimes called an arc or a line.

40

Graphs have numerous applications in many fields.

• Partly as a result of this, graph terminology is

not well standardized!

• Two Examples: What we call a vertex is sometimes called

a node or a point.

What we call an edge is sometimes called an arc or a line.

41

Graphs have numerous applications in many fields.

• Partly as a result of this, graph terminology is

not well standardized!

• Two Examples: What we call a vertex is sometimes called

a node or a point. What we call an edge is sometimes called an arc or a line.

42

Graphs have numerous applications in many fields.

• Partly as a result of this, graph terminology is

not well standardized!

• Two Examples: What we call a vertex is sometimes called

a node or a point. What we call an edge is sometimes called an arc or a line.

• Rosen gives many examples of how graphs are used, some of

which will be highlighted on the following slides.

• Some Terminology:

43

Graphs have numerous applications in many fields.

• Partly as a result of this, graph terminology is

not well standardized!

• Two Examples: What we call a vertex is sometimes called

a node or a point. What we call an edge is sometimes called an arc or a line.

• Rosen gives many examples of how graphs are used, some of

which will be highlighted on the following slides.

• Some Terminology:
• An undirected edge with endpoints u and v is also called:
• An edge that connects/joins u and/to v (or vice versa).
• An edge {u, v} or {v, u} or uv or vu.
• A directed edge whose initial and terminal endpoints are

respectively u and v is also called:

• An edge from u to v.
• An edge that starts at u and ends at v.
• An edge (u, v) or uv.
• (v, u) ≠ (u, v); a directed edge vu ≠ a directed edge uv.

44

Graphs have numerous applications in many fields.

• Partly as a result of this, graph terminology is

not well standardized!

• Two Examples: What we call a vertex is sometimes called

a node or a point. What we call an edge is sometimes called an arc or a line.

• Rosen gives many examples of how graphs are used, some of

which will be highlighted on the following slides.

• Some Terminology:
• An undirected edge with endpoints u and v is also called:
• An edge that connects/joins u and/to v (or vice versa).
• An edge {u, v} or {v, u} or uv or vu.
• A directed edge whose initial and terminal endpoints are

respectively u and v is also called:

• An edge from u to v.
• An edge that starts at u and ends at v.
• An edge (u, v) or uv.
• (v, u) ≠ (u, v); a directed edge vu ≠ a directed edge uv.

45

Graphs have numerous applications in many fields.

• Partly as a result of this, graph terminology is

not well standardized!

• Two Examples: What we call a vertex is sometimes called

a node or a point. What we call an edge is sometimes called an arc or a line.

• Rosen gives many examples of how graphs are used, some of

which will be highlighted on the following slides.

• Some Terminology:
• An undirected edge with endpoints u and v is also called:
• An edge that connects/joins u and/to v (or vice versa).
• An edge {u, v} or {v, u} or uv or vu.
• A directed edge whose initial and terminal endpoints are

respectively u and v is also called:

• An edge from u to v.
• An edge that starts at u and ends at v.
• An edge (u, v) or uv.
• (v, u) ≠ (u, v); a directed edge vu ≠ a directed edge uv.

46

Graphs have numerous applications in many fields.

• Partly as a result of this, graph terminology is

not well standardized!

• Two Examples: What we call a vertex is sometimes called

a node or a point. What we call an edge is sometimes called an arc or a line.

• Rosen gives many examples of how graphs are used, some of

which will be highlighted on the following slides.

• Some Terminology:
• An undirected edge with endpoints u and v is also called:
• An edge that connects/joins u and/to v (or vice versa).
• An edge {u, v} or {v, u} or uv or vu.
• A directed edge whose initial and terminal endpoints are

respectively u and v is also called:

• An edge from u to v.
• An edge that starts at u and ends at v.
• An edge (u, v) or uv.
• (v, u) ≠ (u, v); a directed edge vu ≠ a directed edge uv.

47

Graphs have numerous applications in many fields.

• Partly as a result of this, graph terminology is

not well standardized!

• Two Examples: What we call a vertex is sometimes called

a node or a point. What we call an edge is sometimes called an arc or a line.

• Rosen gives many examples of how graphs are used, some of

which will be highlighted on the following slides.

• Some Terminology:
• An undirected edge with endpoints u and v is also called:
• An edge that connects/joins u and/to v (or vice versa).
• An edge {u, v} or {v, u} or uv or vu.
• A directed edge whose initial and terminal endpoints are

respectively u and v is also called:

• An edge from u to v.
• An edge that starts at u and ends at v.
• An edge (u, v) or uv.
• (v, u) ≠ (u, v); a directed edge vu ≠ a directed edge uv.

48

Graphs have numerous applications in many fields.

• Partly as a result of this, graph terminology is

not well standardized!

• Two Examples: What we call a vertex is sometimes called

a node or a point. What we call an edge is sometimes called an arc or a line.

• Rosen gives many examples of how graphs are used, some of

which will be highlighted on the following slides.

• Some Terminology:
• An undirected edge with endpoints u and v is also called:
• An edge that connects/joins u and/to v (or vice versa).
• An edge {u, v} or {v, u} or uv or vu.
• A directed edge whose initial and terminal endpoints are

respectively u and v is also called:

• An edge from u to v.
• An edge that starts at u and ends at v.
• An edge (u, v) or uv.
• (v, u) ≠ (u, v); a directed edge vu ≠ a directed edge uv.

49

Graphs have numerous applications in many fields.

• Partly as a result of this, graph terminology is

not well standardized!

• Two Examples: What we call a vertex is sometimes called

a node or a point. What we call an edge is sometimes called an arc or a line.

• Rosen gives many examples of how graphs are used, some of

which will be highlighted on the following slides.

• Some Terminology:
• An undirected edge with endpoints u and v is also called:
• An edge that connects/joins u and/to v (or vice versa).
• An edge {u, v} or {v, u} or uv or vu.
• A directed edge whose initial and terminal endpoints are

respectively u and v is also called:

• An edge from u to v.
• An edge that starts at u and ends at v.
• An edge (u, v) or uv.
• (v, u) ≠ (u, v); a directed edge vu ≠ a directed edge uv.

50

Graphs have numerous applications in many fields.

• Partly as a result of this, graph terminology is

not well standardized!

• Two Examples: What we call a vertex is sometimes called

a node or a point. What we call an edge is sometimes called an arc or a line.

• Rosen gives many examples of how graphs are used, some of

which will be highlighted on the following slides.

• Some Terminology:
• An undirected edge with endpoints u and v is also called:
• An edge that connects/joins u and/to v (or vice versa).
• An edge {u, v} or {v, u} or uv or vu.
• A directed edge whose initial and terminal endpoints are

respectively u and v is also called:

• An edge from u to v.
• An edge that starts at u and ends at v.
• An edge (u, v) or uv.
• (v, u) ≠ (u, v); a directed edge vu ≠ a directed edge uv.

51

Graphs have numerous applications in many fields.

• Partly as a result of this, graph terminology is

not well standardized!

• Two Examples: What we call a vertex is sometimes called

a node or a point. What we call an edge is sometimes called an arc or a line.

• Rosen gives many examples of how graphs are used, some of

which will be highlighted on the following slides.

• Some Terminology:
• An undirected edge with endpoints u and v is also called:
• An edge that connects/joins u and/to v (or vice versa).
• An edge {u, v} or {v, u} or uv or vu.
• A directed edge whose initial and terminal endpoints are

respectively u and v is also called:

• An edge from u to v.
• An edge that starts at u and ends at v.
• An edge (u, v) or uv.
• (v, u) ≠ (u, v); a directed edge vu ≠ a directed edge uv.

52

Graphs have numerous applications in many fields.

• Partly as a result of this, graph terminology is

not well standardized!

• Two Examples: What we call a vertex is sometimes called

a node or a point. What we call an edge is sometimes called an arc or a line.

• Rosen gives many examples of how graphs are used, some of

which will be highlighted on the following slides.

• Some Terminology:
• An undirected edge with endpoints u and v is also called:
• An edge that connects/joins u and/to v (or vice versa).
• An edge {u, v} or {v, u} or uv or vu.
• A directed edge whose initial and terminal endpoints are

respectively u and v is also called:

• An edge from u to v.
• An edge that starts at u and ends at v.
• An edge (u, v) or uv.
• (v, u) ≠ (u, v); a directed edge vu ≠ a directed edge uv.

53

From p. 677 of Rosen (8th ed.):

54

From p. 677 of Rosen (8th ed.):

55

From p. 677 of Rosen (8th ed.):

56

From p. 677 of Rosen (8th ed.):

57

From p. 677 of Rosen (8th ed.):

58

From p. 678 of Rosen (8th ed.):

59

From p. 678 of Rosen (8th ed.):

60

From p. 678 of Rosen (8th ed.):

61

From pp. 679 - 80 of Rosen (8th ed.):