# The Mathematics of Networks (Chapter 7)

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1 The Mathematics of Networks (Chapter 7) We have studied how to visit all the edges of a graph (via an Euler path or circuit) and how to visit all the vertices (via a Hamilton circuit).

2 The Mathematics of Networks (Chapter 7) We have studied how to visit all the edges of a graph (via an Euler path or circuit) and how to visit all the vertices (via a Hamilton circuit). What if we just want to connect all the vertices together into a network?

3 The Mathematics of Networks (Chapter 7) We have studied how to visit all the edges of a graph (via an Euler path or circuit) and how to visit all the vertices (via a Hamilton circuit). What if we just want to connect all the vertices together into a network? In other words, What if we just want to connect all the vertices together in a network?

4 The Mathematics of Networks (Chapter 7) We have studied how to visit all the edges of a graph (via an Euler path or circuit) and how to visit all the vertices (via a Hamilton circuit). What if we just want to connect all the vertices together into a network? In other words, What if we just want to connect all the vertices together in a network? Roads, railroads Telephone lines Fiber-optic cable

5 Darwin(DA) Kununurra(KU) Cairns (CS) Broome (BM) Mount Isa (MI) Mackay (MK) Alice Springs(AS) Uluru(UL) Brisbane (BR) Perth (PE) Adelaide (AD) Sydney (SY) Albany(AL) Melbourne (ML) Canberra(CN) Hobart (HO)

6 Darwin(DA) Kununurra(KU) Cairns (CS) Broome (BM) Mount Isa (MI) Mackay (MK) Alice Springs(AS) Uluru(UL) Brisbane (BR) Perth (PE) Adelaide (AD) Sydney (SY) Albany(AL) Melbourne (ML) Canberra(CN) Hobart (HO)

7 Darwin(DA) Kununurra(KU) Cairns (CS) Broome (BM) Mount Isa (MI) Mackay (MK) Alice Springs(AS) Uluru(UL) Brisbane (BR) Perth (PE) Adelaide (AD) Sydney (SY) Albany(AL) Melbourne (ML) Canberra(CN) Too many edges! Hobart (HO)

8 Darwin(DA) Kununurra(KU) Cairns (CS) Broome (BM) Mount Isa (MI) Mackay (MK) Alice Springs(AS) Uluru(UL) Brisbane (BR) Perth (PE) Adelaide (AD) Sydney (SY) Albany(AL) Melbourne (ML) Canberra(CN) Still too many edges Hobart (HO)

9 Darwin(DA) Kununurra(KU) Cairns (CS) Broome (BM) Mount Isa (MI) Mackay (MK) Alice Springs(AS) Uluru(UL) Brisbane (BR) Perth (PE) Adelaide (AD) Sydney (SY) Albany(AL) Melbourne (ML) Canberra(CN) Just right Hobart (HO)

10 Darwin(DA) Kununurra(KU) Cairns (CS) Broome (BM) Mount Isa (MI) Mackay (MK) Alice Springs(AS) Uluru(UL) Brisbane (BR) Perth (PE) Adelaide (AD) Sydney (SY) Albany(AL) Melbourne (ML) Canberra(CN) Another possibility Hobart (HO)

11 Darwin(DA) Kununurra(KU) Cairns (CS) Broome (BM) Mount Isa (MI) Mackay (MK) Alice Springs(AS) Uluru(UL) Brisbane (BR) Perth (PE) Adelaide (AD) Sydney (SY) Albany(AL) Melbourne (ML) Canberra(CN) Not enough edges Hobart (HO)

12 Networks and Spanning Trees Definition: A network is a connected graph. Definition: A spanning tree of a network is a subgraph that 1. connects all the vertices together; and 2. contains no circuits. In graph theory terms, a spanning tree is a subgraph that is both connected and acyclic.

13 Darwin(DA) Kununurra(KU) Cairns (CS) Broome (BM) Mount Isa (MI) Mackay (MK) Alice Springs(AS) Uluru(UL) Brisbane (BR) Perth (PE) Adelaide (AD) Sydney (SY) Albany(AL) Melbourne (ML) Canberra(CN) A spanning tree Hobart (HO)

14 Darwin(DA) Kununurra(KU) Cairns (CS) Broome (BM) Mount Isa (MI) Mackay (MK) Alice Springs(AS) Uluru(UL) Brisbane (BR) Perth (PE) Adelaide (AD) Sydney (SY) Albany(AL) Melbourne (ML) Canberra(CN) A spanning tree Hobart (HO)

15 Darwin(DA) Kununurra(KU) Cairns (CS) Broome (BM) Mount Isa (MI) Mackay (MK) Alice Springs(AS) Uluru(UL) Brisbane (BR) Perth (PE) Adelaide (AD) Sydney (SY) Albany(AL) Melbourne (ML) Canberra(CN) Not a spanning tree (not connected) Hobart (HO)

16 Darwin(DA) Kununurra(KU) Cairns (CS) Broome (BM) Mount Isa (MI) Mackay (MK) Alice Springs(AS) Uluru(UL) Brisbane (BR) Perth (PE) Adelaide (AD) Sydney (SY) Albany(AL) Melbourne (ML) Canberra(CN) Not a spanning tree (connected, but has a circuit) Hobart (HO)

17 Darwin(DA) Kununurra(KU) Cairns (CS) Broome (BM) Mount Isa (MI) Mackay (MK) Alice Springs(AS) Uluru(UL) Brisbane (BR) Perth (PE) Adelaide (AD) Sydney (SY) Albany(AL) Melbourne (ML) Canberra(CN) Not a spanning tree (connected, but has a circuit) Hobart (HO)

18 The Number of Edges in a Spanning Tree In a network with N vertices, how many edges does a spanning tree have?

19 The Number of Edges in a Spanning Tree Imagine starting with N isolated vertices and adding edges one at a time.

20 The Number of Edges in a Spanning Tree Imagine starting with N isolated vertices and adding edges one at a time. Each time you add an edge, you either connect two components together, or close a circuit

21 The Number of Edges in a Spanning Tree Imagine starting with N isolated vertices and adding edges one at a time. Each time you add an edge, you either connect two components together, or close a circuit

22 The Number of Edges in a Spanning Tree Imagine starting with N isolated vertices and adding edges one at a time. Each time you add an edge, you either connect two components together, or close a circuit Stop when the graph is connected (i.e., has only one component).

23 The Number of Edges in a Spanning Tree Imagine starting with N isolated vertices and adding edges one at a time. Each time you add an edge, you either connect two components together, or close a circuit Stop when the graph is connected (i.e., has only one component). You have added exactly N 1 edges.

24 The Number of Edges in a Spanning Tree Imagine starting with N isolated vertices and adding edges one at a time. Each time you add an edge, you either connect two components together, or close a circuit Stop when the graph is connected (i.e., has only one component). You have added exactly N 1 edges. In a network with N vertices, every spanning tree has exactly N 1 edges.

25 The Number of Edges in a Spanning Tree In a network with N vertices, every spanning tree has exactly N 1 edges.

26 The Number of Edges in a Spanning Tree In a network with N vertices, every spanning tree has exactly N 1 edges. Must every set of N 1 edges form a spanning tree?

27 The Number of Edges in a Spanning Tree Answer: No. For example, suppose the network is K 4. Spanning tree Spanning tree Not a spanning tree

28 Spanning Trees in K 2 and K 3 K K

29 Spanning Trees in K

30 Facts about Spanning Trees Suppose we have a network with N vertices. 1. Every spanning tree has exactly N 1 edges.

31 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 the vertices, so it is a spanning tree.

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

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

34 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 The number M N + 1 is called the redundancy of the network, denoted by R. 5. If R = 0, then the network is itself a tree. If R > 0, then there are usually several spanning trees.

35 Counting Spanning Trees We now know that every spanning tree of an N-vertex network has exactly N 1 edges.

36 Counting Spanning Trees We now know that every spanning tree of an N-vertex network has exactly N 1 edges. How many different spanning trees are there?

37 Counting Spanning Trees We now know that every spanning tree of an N-vertex network has exactly N 1 edges. How many different spanning trees are there? Of course, this answer depends on the network itself.

38 Loops and Bridges If an edge of a network is a loop, then it is not in any spanning tree.

39 Loops and Bridges If an edge of a network is a loop, then it is not in any spanning tree. If an edge of a network is a bridge, then it must belong to every spanning tree.

40 Loops and Bridges If an edge of a network is a bridge, then it must belong to every spanning tree.

41 Loops and Bridges If an edge of a network is a bridge, then it must belong to every spanning tree. bridge

42 Loops and Bridges If an edge of a network is a bridge, then it must belong to every spanning tree.

43 Loops and Bridges If an edge of a network is a bridge, then it must belong to every spanning tree.

44 Loops and Bridges If an edge of a network is a bridge, then it must belong to every spanning tree.

45 Loops and Bridges If an edge of a network is a bridge, then it must belong to every spanning tree.

46 Counting Spanning Trees We now know that every spanning tree of an N-vertex network has exactly N 1 edges.

47 Counting Spanning Trees We now know that every spanning tree of an N-vertex network has exactly N 1 edges. How many different spanning trees are there?

48 Counting Spanning Trees We now know that every spanning tree of an N-vertex network has exactly N 1 edges. How many different spanning trees are there? Of course, this answer depends on the network itself.

49 Counting Spanning Trees

50 Counting Spanning Trees

51 Counting Spanning Trees

52 Counting Spanning Trees

53 Counting Spanning Trees

54 Counting Spanning Trees

55 Counting Spanning Trees 3 spanning trees x 3 spanning trees of this triangle of this triangle = 9 total spanning trees

56 Counting Spanning Trees How many spanning trees does this network have?

57 Counting Spanning Trees bridges How many spanning trees does this network have?

58 Counting Spanning Trees 4 bridges 3 How many spanning trees does this network have? 3 4

59 Counting Spanning Trees 4 bridges 3 How many spanning trees does this network have? 3 4 Answer: 4 x 3 x 3 x 4 = 144.

60 Counting Spanning Trees If the graph has circuits that overlap, it is trickier to count spanning trees. For example:

61 Counting Spanning Trees If the graph has circuits that overlap, it is trickier to count spanning trees. For example: There are N = 4 vertices = every spanning tree has N 1 = 3 edges.

62 Counting Spanning Trees If the graph has circuits that overlap, it is trickier to count spanning trees. For example: There are N = 4 vertices = every spanning tree has N 1 = 3 edges. List all the sets of three edges and cross out the ones that are not spanning trees.

63 Counting Spanning Trees ways to select three edges

64 Counting Spanning Trees ways to select three edges 2 are not trees Total: 8 spanning trees

65 The Number of Spanning Trees of K N (Not in Tannenbaum!) Since K N has N vertices, we know that every spanning tree of K N has N 1 edges. How many different spanning trees are there?

66 The Number of Spanning Trees of K N (Not in Tannenbaum!) Since K N has N vertices, we know that every spanning tree of K N has N 1 edges. How many different spanning trees are there? We have already seen the answers for K 2, K 3, and K 4.

67 The Number of Spanning Trees of K N Number of vertices (N) Number of spanning trees in K N

68 The Number of Spanning Trees of K N Number of vertices (N) Number of spanning trees in K N

69 The Number of Spanning Trees of K N Number of vertices (N) Number of spanning trees in K N

70 The Number of Spanning Trees of K N Number of vertices (N) Number of spanning trees in K N

71 The Number of Spanning Trees of K N Number of vertices (N) Number of spanning trees in K N

72 The Number of Spanning Trees of K N Number of vertices (N) Number of spanning trees in K N What s the pattern?

73 The Number of Spanning Trees of K N Number of vertices (N) Number of spanning trees in K N

74 The Number of Spanning Trees of K N Number of vertices (N) Number of spanning trees in K N =

75 The Number of Spanning Trees of K N Number of vertices (N) Number of spanning trees in K N = =

76 The Number of Spanning Trees of K N Number of vertices (N) Number of spanning trees in K N = = =

77 The Number of Spanning Trees of K N Number of vertices (N) Number of spanning trees in K N = = = =

78 The Number of Spanning Trees of K N Number of vertices (N) Number of spanning trees in K N = = = = =

79 The Number of Spanning Trees of K N Number of vertices (N) Number of spanning trees in K N = = = = = = 8 6

80 The Number of Spanning Trees of K N Number of vertices (N) Number of spanning trees in K N = = = = = = = 8 6

81 The Number of Spanning Trees of K N Number of vertices (N) Number of spanning trees in K N 1 1 = = = = = = = = 8 6

82 The Number of Spanning Trees of K N Number of vertices (N) Number of spanning trees in K N 1 1 = = = = = = = = 8 6 Cayley s Formula: The number of spanning trees in K N is N N 2.

83 Cayley s Formula: The number of spanning trees in K N is N N 2. For example, K 16 (the Australia graph!) has = 72, 057, 594, 037, 927, 936 spanning trees. (By comparison, the number of Hamilton circuits is only 15! = 1, 307, 674, 368, 000.)

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