The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe...
Transcript of The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe...
![Page 1: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/1.jpg)
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
![Page 2: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/2.jpg)
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
![Page 3: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/3.jpg)
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
![Page 4: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/4.jpg)
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
![Page 5: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/5.jpg)
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)
![Page 6: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/6.jpg)
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)
![Page 7: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/7.jpg)
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)
![Page 8: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/8.jpg)
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)
![Page 9: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/9.jpg)
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)
![Page 10: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/10.jpg)
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)
![Page 11: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/11.jpg)
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)
![Page 12: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/12.jpg)
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.
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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)
![Page 14: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/14.jpg)
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)
![Page 15: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/15.jpg)
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)
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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)
![Page 17: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/17.jpg)
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)
![Page 18: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/18.jpg)
The Number of Edges in a Spanning Tree
In a network with N vertices, how many edges does aspanning tree have?
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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.
![Page 20: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/20.jpg)
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.
![Page 21: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/21.jpg)
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.
![Page 22: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/22.jpg)
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.
![Page 23: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/23.jpg)
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.
![Page 24: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/24.jpg)
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.
![Page 25: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/25.jpg)
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?
![Page 26: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/26.jpg)
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?
![Page 27: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/27.jpg)
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
![Page 28: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/28.jpg)
Spanning Trees in K2 and K3
K3
K2
32
1
32
1
32
1
1 2
![Page 29: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/29.jpg)
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
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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.
![Page 31: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/31.jpg)
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.
![Page 32: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/32.jpg)
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.
![Page 33: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/33.jpg)
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.
![Page 34: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/34.jpg)
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.
![Page 35: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/35.jpg)
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.
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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.
![Page 37: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/37.jpg)
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.
![Page 38: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/38.jpg)
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.
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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.
![Page 40: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/40.jpg)
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.
![Page 41: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/41.jpg)
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
![Page 42: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/42.jpg)
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.
![Page 43: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/43.jpg)
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.
![Page 44: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/44.jpg)
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.
![Page 45: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/45.jpg)
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.
![Page 46: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/46.jpg)
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.
![Page 47: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/47.jpg)
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.
![Page 48: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/48.jpg)
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.
![Page 49: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/49.jpg)
Counting Spanning Trees
![Page 50: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/50.jpg)
Counting Spanning Trees
![Page 51: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/51.jpg)
Counting Spanning Trees
![Page 52: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/52.jpg)
Counting Spanning Trees
![Page 53: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/53.jpg)
Counting Spanning Trees
![Page 54: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/54.jpg)
Counting Spanning Trees
![Page 55: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/55.jpg)
Counting Spanning Trees
xof this triangleof this triangle
3 spanning trees 3 spanning trees
= 9 total spanning trees
![Page 56: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/56.jpg)
Counting Spanning Trees
How many spanningtrees does thisnetwork have?
![Page 57: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/57.jpg)
Counting Spanning Trees
bridges How many spanningtrees does thisnetwork have?
![Page 58: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/58.jpg)
Counting Spanning Trees
bridges How many spanningtrees does thisnetwork have?
4
4
3
3
![Page 59: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/59.jpg)
Counting Spanning Trees
bridges How many spanningtrees does thisnetwork have?
4
4
3
34 x 3 x 3 x 4 =Answer:
144.
![Page 60: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/60.jpg)
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.
![Page 61: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/61.jpg)
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.
![Page 62: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/62.jpg)
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.
![Page 63: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/63.jpg)
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
![Page 64: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/64.jpg)
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
![Page 65: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/65.jpg)
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.
![Page 66: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/66.jpg)
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.
![Page 67: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/67.jpg)
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?
![Page 68: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/68.jpg)
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?
![Page 69: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/69.jpg)
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?
![Page 70: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/70.jpg)
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?
![Page 71: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/71.jpg)
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?
![Page 72: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/72.jpg)
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?
![Page 73: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/73.jpg)
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.
![Page 74: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/74.jpg)
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.
![Page 75: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/75.jpg)
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.
![Page 76: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/76.jpg)
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.
![Page 77: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/77.jpg)
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.
![Page 78: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/78.jpg)
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.
![Page 79: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/79.jpg)
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.
![Page 80: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/80.jpg)
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.
![Page 81: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/81.jpg)
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.
![Page 82: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/82.jpg)
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.
![Page 83: The Mathematics of Networks (Chapter 7)jlmartin/courses/math105-F11/Lectures/chapter7-part1.pdfThe Number of Edges in a Spanning Tree I Imagine starting with N isolated vertices and](https://reader035.fdocuments.us/reader035/viewer/2022071006/5fc3997e56dd390716727015/html5/thumbnails/83.jpg)
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.)