The Mathematics of Networks (Chapter 7)

We have studied how to visit all the edges of a graph (via anEuler path or circuit) and how to visit all the vertices (via aHamilton circuit).

What if we just want to connect all the verticestogether into a network?

In other words, What if we just want to connect all thevertices together in a network?

I Roads, railroads

I Telephone lines

I Fiber-optic cable

The Mathematics of Networks (Chapter 7)

We have studied how to visit all the edges of a graph (via anEuler path or circuit) and how to visit all the vertices (via aHamilton circuit).

What if we just want to connect all the verticestogether into a network?

In other words, What if we just want to connect all thevertices together in a network?

I Roads, railroads

I Telephone lines

I Fiber-optic cable

The Mathematics of Networks (Chapter 7)

We have studied how to visit all the edges of a graph (via anEuler path or circuit) and how to visit all the vertices (via aHamilton circuit).

What if we just want to connect all the verticestogether into a network?

In other words, What if we just want to connect all thevertices together in a network?

I Roads, railroads

I Telephone lines

I Fiber-optic cable

The Mathematics of Networks (Chapter 7)

We have studied how to visit all the edges of a graph (via anEuler path or circuit) and how to visit all the vertices (via aHamilton circuit).

What if we just want to connect all the verticestogether into a network?

In other words, What if we just want to connect all thevertices together in a network?

I Roads, railroads

I Telephone lines

I Fiber-optic cable

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(UL)Uluru

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)

Too many edges!

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)

Still too many edges

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)

Just right

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)

Another possibility

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)

Not enough edges

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)

Networks and Spanning Trees

Definition: A network is a connected graph.

Definition: A spanning tree of a network is a subgraphthat

1. connects all the vertices together; and

2. contains no circuits.

In graph theory terms, a spanning tree is a subgraph that isboth connected and acyclic.

A spanning tree

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)

A spanning tree

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)

Not a spanning tree

(not connected)

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)

Not a spanning tree

(connected, but has a circuit)

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)

Not a spanning tree

(connected, but has a circuit)

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)

The Number of Edges in a Spanning Tree

In a network with N vertices, how many edges does aspanning tree have?

The Number of Edges in a Spanning Tree

I Imagine starting with N isolated vertices and addingedges one at a time.

I Each time you add an edge, you either

I connect two components together, or

I

I Stop when the graph is connected (i.e., has only onecomponent).

I You have added exactly N − 1 edges.

In a network with N vertices, every spanning tree hasexactly N − 1 edges.

The Number of Edges in a Spanning Tree

I Imagine starting with N isolated vertices and addingedges one at a time.

I Each time you add an edge, you eitherI connect two components together, orI close a circuit

I Stop when the graph is connected (i.e., has only onecomponent).

I You have added exactly N − 1 edges.

In a network with N vertices, every spanning tree hasexactly N − 1 edges.

The Number of Edges in a Spanning Tree

I Imagine starting with N isolated vertices and addingedges one at a time.

I Each time you add an edge, you eitherI connect two components together, orI close a circuit

I Stop when the graph is connected (i.e., has only onecomponent).

I You have added exactly N − 1 edges.

In a network with N vertices, every spanning tree hasexactly N − 1 edges.

The Number of Edges in a Spanning Tree

I Imagine starting with N isolated vertices and addingedges one at a time.

I Each time you add an edge, you eitherI connect two components together, orI close a circuit

I Stop when the graph is connected (i.e., has only onecomponent).

I You have added exactly N − 1 edges.

In a network with N vertices, every spanning tree hasexactly N − 1 edges.

The Number of Edges in a Spanning Tree

I Imagine starting with N isolated vertices and addingedges one at a time.

I Each time you add an edge, you eitherI connect two components together, orI close a circuit

I Stop when the graph is connected (i.e., has only onecomponent).

I You have added exactly N − 1 edges.

In a network with N vertices, every spanning tree hasexactly N − 1 edges.

The Number of Edges in a Spanning Tree

I Imagine starting with N isolated vertices and addingedges one at a time.

I Each time you add an edge, you eitherI connect two components together, orI close a circuit

I Stop when the graph is connected (i.e., has only onecomponent).

I You have added exactly N − 1 edges.

In a network with N vertices, every spanning tree hasexactly N − 1 edges.

The Number of Edges in a Spanning Tree

In a network with N vertices, every spanning tree hasexactly N − 1 edges.

Must every set of N − 1 edges form a spanning tree?

The Number of Edges in a Spanning Tree

In a network with N vertices, every spanning tree hasexactly N − 1 edges.

Must every set of N − 1 edges form a spanning tree?

The Number of Edges in a Spanning Tree

Answer: No.

For example, suppose the network is K4.

Spanning tree Spanning tree Not a spanning tree

Spanning Trees in K2 and K3

K3

K2

32

1

32

1

32

1

1 2

Spanning Trees in K4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

Facts about Spanning Trees

Suppose we have a network with N vertices.

1. Every spanning tree has exactly N − 1 edges.

2. If a set of N − 1 edges is acyclic, then it connects all thevertices, so it is a spanning tree.

3. If a set of N − 1 edges connects all the vertices, then it isacyclic, so it is a spanning tree.

Facts about Spanning Trees

Suppose we have a network with N vertices.

1. Every spanning tree has exactly N − 1 edges.

2. If a set of N − 1 edges is acyclic, then it connects all thevertices, so it is a spanning tree.

3. If a set of N − 1 edges connects all the vertices, then it isacyclic, so it is a spanning tree.

Facts about Spanning Trees

Suppose we have a network with N vertices.

1. Every spanning tree has exactly N − 1 edges.

2. If a set of N − 1 edges is acyclic, then it connects all thevertices, so it is a spanning tree.

3. If a set of N − 1 edges connects all the vertices, then it isacyclic, so it is a spanning tree.

Facts about Spanning Trees

4. In a network with N vertices and M edges,

M ≥ N − 1

(otherwise it couldn’t possibly be connected!) That is,

M − N + 1 ≥ 0.

The number M − N + 1 is called the redundancy of thenetwork, denoted by R .

5. If R = 0, then the network is itself a tree.If R > 0, then there are usually several spanning trees.

Facts about Spanning Trees

4. In a network with N vertices and M edges,

M ≥ N − 1

(otherwise it couldn’t possibly be connected!) That is,

M − N + 1 ≥ 0.

The number M − N + 1 is called the redundancy of thenetwork, denoted by R .

5. If R = 0, then the network is itself a tree.If R > 0, then there are usually several spanning trees.

Counting Spanning Trees

We now know that every spanning tree of an N-vertex networkhas exactly N − 1 edges.

How many different spanning trees are there?

Of course, this answer depends on the network itself.

Counting Spanning Trees

We now know that every spanning tree of an N-vertex networkhas exactly N − 1 edges.

How many different spanning trees are there?

Of course, this answer depends on the network itself.

Counting Spanning Trees

We now know that every spanning tree of an N-vertex networkhas exactly N − 1 edges.

How many different spanning trees are there?

Of course, this answer depends on the network itself.

Loops and Bridges

I If an edge of a network is a loop, then it is not in anyspanning tree.

I If an edge of a network is a bridge, then it must belongto every spanning tree.

Loops and Bridges

I If an edge of a network is a loop, then it is not in anyspanning tree.

I If an edge of a network is a bridge, then it must belongto every spanning tree.

Loops and Bridges

I If an edge of a network is a loop, then it is not in anyspanning tree.

I If an edge of a network is a bridge, then it must belongto every spanning tree.

Loops and Bridges

I If an edge of a network is a loop, then it is not in anyspanning tree.

I If an edge of a network is a bridge, then it must belongto every spanning tree.

bridge

Loops and Bridges

I If an edge of a network is a loop, then it is not in anyspanning tree.

I If an edge of a network is a bridge, then it must belongto every spanning tree.

Loops and Bridges

I If an edge of a network is a loop, then it is not in anyspanning tree.

I If an edge of a network is a bridge, then it must belongto every spanning tree.

Loops and Bridges

I If an edge of a network is a loop, then it is not in anyspanning tree.

I If an edge of a network is a bridge, then it must belongto every spanning tree.

Loops and Bridges

I If an edge of a network is a loop, then it is not in anyspanning tree.

I If an edge of a network is a bridge, then it must belongto every spanning tree.

Counting Spanning Trees

We now know that every spanning tree of an N-vertex networkhas exactly N − 1 edges.

How many different spanning trees are there?

Of course, this answer depends on the network itself.

Counting Spanning Trees

We now know that every spanning tree of an N-vertex networkhas exactly N − 1 edges.

How many different spanning trees are there?

Of course, this answer depends on the network itself.

Counting Spanning Trees

We now know that every spanning tree of an N-vertex networkhas exactly N − 1 edges.

How many different spanning trees are there?

Of course, this answer depends on the network itself.

Counting Spanning Trees

Counting Spanning Trees

Counting Spanning Trees

Counting Spanning Trees

Counting Spanning Trees

Counting Spanning Trees

Counting Spanning Trees

xof this triangleof this triangle

3 spanning trees 3 spanning trees

= 9 total spanning trees

Counting Spanning Trees

How many spanningtrees does thisnetwork have?

Counting Spanning Trees

bridges How many spanningtrees does thisnetwork have?

Counting Spanning Trees

bridges How many spanningtrees does thisnetwork have?

4

4

3

3

Counting Spanning Trees

bridges How many spanningtrees does thisnetwork have?

4

4

3

34 x 3 x 3 x 4 =Answer:

144.

Counting Spanning Trees

If the graph has circuits that overlap, it is trickier to countspanning trees. For example:

1 2

3 4

I There are N = 4 vertices =⇒every spanning tree has N − 1 = 3 edges.

I List all the sets of three edges and cross out the ones thatare not spanning trees.

Counting Spanning Trees

If the graph has circuits that overlap, it is trickier to countspanning trees. For example:

1 2

3 4

I There are N = 4 vertices =⇒every spanning tree has N − 1 = 3 edges.

I List all the sets of three edges and cross out the ones thatare not spanning trees.

Counting Spanning Trees

If the graph has circuits that overlap, it is trickier to countspanning trees. For example:

1 2

3 4

I There are N = 4 vertices =⇒every spanning tree has N − 1 = 3 edges.

I List all the sets of three edges and cross out the ones thatare not spanning trees.

Counting Spanning Trees

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

10 ways to select three edges

Counting Spanning Trees

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

1 2

3 4

2 are not treesTotal: 8 spanning trees

10 ways to select three edges

The Number of Spanning Trees of KN

(Not in Tannenbaum!)

Since KN has N vertices, we know that every spanning tree ofKN has N − 1 edges.

How many different spanning trees are there?

We have already seen the answers for K2, K3, and K4.

The Number of Spanning Trees of KN

(Not in Tannenbaum!)

Since KN has N vertices, we know that every spanning tree ofKN has N − 1 edges.

How many different spanning trees are there?

We have already seen the answers for K2, K3, and K4.

The Number of Spanning Trees of KN

Number of vertices (N) Number of spanning trees in KN

2 13 34 16

What’s the pattern?

The Number of Spanning Trees of KN

Number of vertices (N) Number of spanning trees in KN

1 12 13 34 16

What’s the pattern?

The Number of Spanning Trees of KN

Number of vertices (N) Number of spanning trees in KN

1 12 13 34 165 125

What’s the pattern?

The Number of Spanning Trees of KN

Number of vertices (N) Number of spanning trees in KN

1 12 13 34 165 1256 1296

What’s the pattern?

The Number of Spanning Trees of KN

Number of vertices (N) Number of spanning trees in KN

1 12 13 34 165 1256 12967 168078 262144

What’s the pattern?

The Number of Spanning Trees of KN

Number of vertices (N) Number of spanning trees in KN

1 12 13 34 165 1256 12967 168078 262144

What’s the pattern?

The Number of Spanning Trees of KN

Number of vertices (N) Number of spanning trees in KN

1 12 13 34 165 1256 12967 168078 262144

Cayley’s Formula:The number of spanning trees in KN is NN−2.

The Number of Spanning Trees of KN

Number of vertices (N) Number of spanning trees in KN

1 12 13 34 16 = 42

5 1256 12967 168078 262144

Cayley’s Formula:The number of spanning trees in KN is NN−2.

The Number of Spanning Trees of KN

Number of vertices (N) Number of spanning trees in KN

1 12 13 34 16 = 42

5 125 = 53

6 12967 168078 262144

Cayley’s Formula:The number of spanning trees in KN is NN−2.

The Number of Spanning Trees of KN

Number of vertices (N) Number of spanning trees in KN

1 12 13 3 = 31

4 16 = 42

5 125 = 53

6 12967 168078 262144

Cayley’s Formula:The number of spanning trees in KN is NN−2.

The Number of Spanning Trees of KN

Number of vertices (N) Number of spanning trees in KN

1 12 13 3 = 31

4 16 = 42

5 125 = 53

6 1296 = 64

7 168078 262144

Cayley’s Formula:The number of spanning trees in KN is NN−2.

The Number of Spanning Trees of KN

Number of vertices (N) Number of spanning trees in KN

1 12 13 3 = 31

4 16 = 42

5 125 = 53

6 1296 = 64

7 16807 = 75

8 262144

Cayley’s Formula:The number of spanning trees in KN is NN−2.

The Number of Spanning Trees of KN

Number of vertices (N) Number of spanning trees in KN

1 12 13 3 = 31

4 16 = 42

5 125 = 53

6 1296 = 64

7 16807 = 75

8 262144 = 86

Cayley’s Formula:The number of spanning trees in KN is NN−2.

The Number of Spanning Trees of KN

Number of vertices (N) Number of spanning trees in KN

1 12 1 = 20

3 3 = 31

4 16 = 42

5 125 = 53

6 1296 = 64

7 16807 = 75

8 262144 = 86

Cayley’s Formula:The number of spanning trees in KN is NN−2.

The Number of Spanning Trees of KN

Number of vertices (N) Number of spanning trees in KN

1 1 = 1−1

2 1 = 20

3 3 = 31

4 16 = 42

5 125 = 53

6 1296 = 64

7 16807 = 75

8 262144 = 86

Cayley’s Formula:The number of spanning trees in KN is NN−2.

The Number of Spanning Trees of KN

Number of vertices (N) Number of spanning trees in KN

1 1 = 1−1

2 1 = 20

3 3 = 31

4 16 = 42

5 125 = 53

6 1296 = 64

7 16807 = 75

8 262144 = 86

Cayley’s Formula:The number of spanning trees in KN is NN−2.

Cayley’s Formula:The number of spanning trees in KN is NN−2.

For example, K16 (the Australia graph!) has

1614 = 72, 057, 594, 037, 927, 936

spanning trees.

(By comparison, the number of Hamilton circuits is “only”

15! = 1, 307, 674, 368, 000.)