Example: Graphspot.pcc.edu/~khatwal/Mth_231/Graphtheory.pdf · 1 kavita hatwal math 231 fall 2002 Graph theory a graph is a finite set of dots called vertices (or nodes) connected

1

kavita hatwal math 231 fall 2002

Graph theorya graph is a finite set of dots called vertices (or nodes) connected by links called edges (or arcs). a simple graph is a (usually finite) set of vertices V and set of unordered pairs of distinct elements of V called edges.

Not all graphs are simple. Sometimes a pair of vertices are connected by multiple edge called parallel edges yielding a multigraph. At times vertices are even connected to themselves by an edge called a loop. Finally, edges can also be given a direction yielding a directed graph (or digraph). a multigraph is a graph with multiple edges between the same vertices. Formally: a multigraph is a set V of vertices along, a set E of edges, and a function f from E to {{u,v}|u,v in V; u,v distinct}.

An edge is said to be incident on its endpoints or vertices.Two vertices are adjacent if they are connected by an edge. Two edges incident on the same vertex are called adjacent. A vertex with a loop is adjacent to itself. A vertex on which no edges are incident is called isolated.


Example: Graph

• Let G = (V, E) where– V = {A, B, C, D, E},– E = {{A, B}, {A, C}, {A, C}, {B, B}, {B, C},

{D, E}}.

D

AB

C

E

2


Terminology

• The edge {a, b} has endpoints a and b.• The edge {a, b} connects the endpoints a

and b.• Vertices that are connected by an edge are

adjacent.


Directed Graphs

• A directed graph G is a pair (V, E) where– V is a set of vertices,– E is a multiset of ordered pairs of vertices.

• That is, each edge has a direction.• A directed graph is also called a digraph.

3


Example: Directed Graph

• Let G = (V, E) where– V = {A, B, C, D, E},– E = {(A, B), (A, C), (C, A), (B, B), (B, C),

(D, E)).A

B

C D

E


Simple Graphs

• A simple graph G is a pair (V, E) where– V is a set of vertices,– E is a set of edges,

where each edge is a set of vertices.

4


A graph with no vertices is called empty, and a one with at least one vertex is called non empty.

Page604, 11.1.1, 11.1.2Page608, 11.1.6,

A complete graph with n vertices (denoted Kn) is a graph with n vertices in which each vertex is connected to each of the others (with one edge between each pair of vertices).

A graph is bipartite if its vertices can be partitioned into two disjoint subsets U and V such that each edge connects a vertex from U to one from V. A bipartite graph is a complete bipartite graph if every vertex in U is connected to every vertex in V. If U has n elements and V has m, then we denote the resulting complete bipartite graph by Kn,m. Page610, blue box


Example: Bipartite Graph

5


Example: Complete Bipartite Graph

K3, 4


Subgraphs

• A graph H = (V(H), E(H)) is a subgraph of a graph G = (V(G), E(G)) if V(H) ⊆ V(G) and E(H) ⊆ E(G).

6


A

B

CD

EF

Example: Subgraph

G = {A, B, C, D, E, F}

H = {A, C, D, F}


Degree of a Vertex

• The degree of a vertex v is the number of edges that include v.– If the vertex has a loop, it is counted twice.

• In a directed graph,– The in-degree of v is the number of edges

directed into v.– The out-degree of v is the number of edges

directed out of v.

7


Degree of a Vertex

• The total degree of a graph G is the sum of the degrees of its vertices.


Example: Degree

A

B

CD

EF

Adeg(A) = 3

8


The total degree of a graph is • 2 * (the number of edges in G)• is even.

Corollary to the Handshaking Theorem:

An undirected graph has an even number of vertices with an odd degree.

Handshaking Theorem


Is it possible to draw an undirected graph of 5 vertices with the following properties: deg(v1) = 5, deg(v2) = 2, deg(v3) = 2, deg(v4) = 1, deg(v5) = 3 ?

• Answer (Version#1):Total degree = 5 + 2 + 2 + 1 + 3

= 13 (odd!!!)…violates handshaking theorem. Therefore, such a graph is not possible.

9


Is it possible to draw an undirected graph of 5 vertices with the following properties: deg(v1) = 5, deg(v2) = 2, deg(v3) = 2, deg(v4) = 1, deg(v5) = 3 ?

• Answer: (Version#2: using corollary)How many vertices are there with odd degree? v1, v4, v5=> 3 of them! But there must be an even number of odd degree vertices. Therefore such a graph is not possible to draw.


Given an undirected graph with 5 vertices and 7 edges, and that deg(v1) = 3, deg(v2) = 1, deg(v3) = 5, deg(v4) = 2, what is the degree of v5?

• Answer:Total degree = 3+1+5+2 + deg(v5) = 2x7

(by handshaking theorem)Therefore deg(v5) = 3

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In a group of 15 people, is it possible for each person to have exactly 3 friends?

• Answer:The problem can be reduced to a graph problem. Each person is a vertex, therefore there are 15 vertices. We connect an edge between 2 vertices if they are friends.If a person has 3 friends, then every vertex has a degree of 3. Is such a graph possible?

deg(v)v∈V

= 15 x 3 = 45 (odd!) Therefore, not possible.


The Seven Bridges of Königsberg

• Is it possible to traverse each bridge exactly once and end up where you started?

?

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• Represent each landmass as a vertex.Represent each bridge between landmasses as an edge between vertices.


• Let G be a graph with vertices v and w.• A walk from v to w is a finite alternating sequence of

vertices and edgesv0e1v1e2v2…vn – 1en – 1vn

where v0 = v, vn = w, and each ei has endpoints vi – 1and vi.

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• A path from v to w is a walk from v to w in which the edges are distinct.

• A simple path from v to w is a path from v to w in which the vertices are distinct.

• A closed walk is a walk that starts and ends at the same vertex.


• A circuit is a closed path.• A simple circuit is a circuit in which the

vertices are distinct, except for the first and last vertices.

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Page 622, table.Page621, 11.2.1, 11.2.2

A graph is connected if there is a path connecting every pair of vertices. A graph that is not connected can be divided into connected components (disjoint connected subgraphs). For example, this graph is made of three connected components.Page625, lemma 11.2.1, 11.2.2


Example: A Walk

• A walk from A to F.

A

B

C

D

E

F

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Example: A Path

• A path from A to F.

A

B

C

D

E

F


Example: A Simple Path

• A simple path from A to F.

A

B

C

D

E

F

15


Example: A Closed Walk

• A closed walk from A to A.

A

B

C

D

E

F


Example: A Circuit

• A circuit from A to A.

A

B

C

D

E

F

16


Example: A Simple Circuit

• A simple circuit from A to A.

A

B

C

D

E

F


Example: Connected Components

17


Connected Graphs

• Two vertices v and w of a graph G are connectedif there is a walk from v to w.

• A graph G is connected if every pair of vertices of G are connected.

• A subgraph H of a graph G is a connected component of G if– H is connected, and– There is no connected subgraph K of G of which H is a

proper subgraph.


Example: Connected Components

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Euler Circuits

• An Euler circuit of a graph G is a circuit that contains every vertex of G at least once and every edge of G exactly once.


Example: An Euler Circuit

19


A

B

CD

EF

Example: Subgraph

G = {A, B, C, D, E, F}

H = {A, C, D, F}


Origin of ProblemIn the town of Konigsberg there was an island in the river connected by seven bridges to the rest of the city. The question was whether you could find a path starting and ending at the same spot, that crossed each bridge exactly once.Solution of ProblemThis problem was solved by Euler, who represented each bridge as an edge, and each terminal point of a bridge as a node in a graph. Then he found a necessary and sufficient condition on each of the nodes: each node must have even degree (even number of edges incident to it). (page 620). All nodes have odd degree. So it is impossible to find a closed path that uses each edge exactly once. Euler's TheoremLet G be a connected graph (every pair of nodes is joined by some path). Then a necessary and sufficient condition for G to have an Euler circuit is that each node of G has even degree.So if a graph has an euler circuit, then every vertex of the graph has even degree.If some vertex of a graph has odd degree, then the graph does not have an Euler circuit.

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Hamilton CircuitsA seemingly related problem is the Hamiltonian circuit problem. Here, goal is to traverse the entire graph, starting at node a and visiting each node exactly once, before returning to a. Notice this means you cannot traverse a given edge more than once (Exception: G has only two nodes), for then you would be visiting at least one of the endpoints of that edge more than once. So in this sense, the Hamilton Circuit problem is more demanding than the Euler circuit problem.


Any edge may be omitted, as long as its endpoints can be reached via some other path.

No edges may be omitted (by defn of the problem).

No node may be omitted (by defn of the problem).

No node may be omitted (if so, one of the edges incident to the node would not be traversed).

Repeated traversals of a given edge are not permitted, since that would result in visiting a node more than once.

Traverse each edge exactly once (by defn of problem).

Visit each node exactly once.Repeated visits to a given node are allowed.

Hamilton CircuitEuler Circuit

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Unlike the Euler circuit problem, finding Hamilton circuits is hard. There is no simple set of necessary and sufficient conditions, and no simple algorithm. But here is a list of conditions that must be satisfied:

Necessary conditions for Hamilton circuitAssume G has more than 2 nodes. 1. No vertex of degree 1

If a has degree 1, then the other endpoint of the edge incident to a must be visited at least twice in any circuit of G.

2. If a node has degree 2, then both edges incident to it must be in any Hamilton circuit.

3. No smaller circuits contained in any Hamilton circuit (the start/endpoint of any smaller circuit would have to be visited twice).

4. There must exist a subgraph H of G with the following properties: 5. H contains every vertex of G 6. H is connected 7. H has the same number of edges as vertices 8. H has every node with degree 2 9. The subgraph H is the Hamilton circuit in the graph G. Page 634 Prop 11.2.6


Example: Hamiltonian Circuit

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Travelling Salesperson ProblemPage635, 11.2.9This is a famous problem. It goes like this: 1. A salesperson has a route among n cities. 2. He/she would like to visit all the cities, but in a way to minimize the total

distance travelled. This is a weighted Hamilton circuit problem. You have to find a route that visits

each city exactly once, but in addition, the edges connecting the cities are weighted according to the distances between those cities. So you want the sum of the edge weights to be minimized. Graph representation of problem

The problem translates into finding the minimum weighted Hamiltonian circuit in a connected graph on n vertices. It is difficult because:

1. There is no simple algorithm for finding a hamiltonian circuit. 2. Among hamiltonian circuits, the best you can do is compute the "weight"

(summed distance) of each, and find the min. For a complete graph on n nodes, there are n! different hamiltonian circuits.

Comparing them all is beyond the range of any known (or possible) computer, when n gets large.


Matrix representations of graphs

What is a matrix?An a (read “m by n”)matrix over a set A is a rectangular array of elements of S arranged into m rows and n columns.

nm×

=

mnmjmm

inijii

nj

nj

aaaaaaaa

aaaaaaaa

A

............

......

......

21

21

222221

111211

MMMMi’th row of A

j’th column of A

ijwe a A write =

23


]...[isA matrix of rowth i' The

21 inii aaa

mj

2j

1j

a

aa

Amatrix ofcolumn th j' The

M


Equal MatrixTwo matrices A and B are said to be equal , if both A and B have the same size and the corresponding entries are equal.That is,aij=bij for all i = 1,2,…,n and j = 1,2,…, nSquare matrixIf a matrix A has n rows and n columns then we say it's a square matrix. In a square matrix the elements ai,i , with i = 1,2,3,... , are called diagonal elements.Diagonal matrixA diagonal matrix is a square matrix with all non-diagonal elements 0.The diagonal matrix is completely denoted by the diagonal elements.Example. [7 0 0] [0 5 0] [0 0 6] The matrix is denoted by diag(7 , 5 , 6) An identity matrix IAn identity matrix I is a diagonal matrix with all diagonal element = 1. The tranpose of a matrixThe n x m matrix A' is the transpose of the m x n matrix A if and only if The ith row of A = the ith column of A' for (i = 1,2,3,..n)So ai,j = aj,i'

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A symmetric matrixA square matrix is called symmetric if it is equal to its transpose.Then ai,j = aj,i , for all i and j. Page 641 #11.3.1Multiplication of a row matrix by a column matrixThis multiplication is only possible if the row matrix and the column matrix have the same number of elements. The result is a ordinary number ( 1 x 1 matrix).To multiply the row by the column, one multiplies corresponding elements, then adds the results.Example. [2 1 3] [1]

[2] = [19] [5]

Multiplication of two matrices A.BThis product is defined only if A is a (l x m) matrix and B is a (m x n) matrix.So the number of columns of A has to be equal to the number of rows of B.The product C = A.B then is a (l x n) matrix.The element of the ith row and the jth column of the product is found by multiplying the ith row of A by the jth column of B. ci,j = sumk (ai,k.bk,j)


Example.

Is matrix multiplication commutative?

But matrix multiplication is associative. That is A(BC) = (AB)CDefinitionFor any square matrix A, the powers of A are defined as follows:

Where I is the identity matrix

for all integers n 1

=

8475

2231

1221

=

−−

????

2222

1111

=

????

1221

2231

1

0

−=

=nn AAA

IAnn ×

≥

25


Matrices and directed graphs

Graphs can also be represented in the form of matrices. The major advantage of matrix representation is that the calculation of paths and cycles can easily be performed using well known operations of matrices. However, the disadvantage is that this form of representation takes away from the visual aspect of graphs. It would be difficult to illustrate in a matrix, properties that are easily illustrated graphically.

Page 641, figure 11.3.1

An adjacency matrix is defined as follows: Let G be a graph with "n" vertices that are assumed to be ordered from v1 to vn. The n x n matrix A, in which

aij= 1 if there exists a path from vi to vjaij = 0 otherwise

is called an adjacency matrix.

• directed graphs and equivalence relations of a binary relation.


Example: Adjacency Matrices

• Let the graph be

The adjacency matrix is

0100110101010100010111010

A B

CD

E

26


• Theorem: Let G be a simple graph. Then– The adjacency matrix of G is symmetric.– The main diagonal is 0.– The degree of vertex vi is the sum of the

elements of row i (or column i).


• The adjacency matrix is

0100110101010100010111010

The row totals are 3, 2, 2, 3, 2, which are the same as the degrees of the vertices.

27


• Theorem: Let G be a simple digraph and let A be the adjacency matrix of G. Then– The indegree of vi equals the sum of column i.– The outdegree of vi equals the sum of row i.

• Matrices and connected components.Page 645


Counting Walks

• Theorem: Let G be a graph and let A be the adjacency matrix of G. Then the (i, j)-element of An equals the number of walks of length n from vi to vj.

• The existence of an edge between two vertices vi and vjis shown by an entry of 1 in the ith row and jth column of the adjacency matrix. This entry represents a walk of length 1 from vi to vj.

• To compute a walk of length 2, the matrix of length 1 must be multiplied by itself, and the product matrix is the matrix representation of path of length 2.

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Example: Number of Walks

• Let G be the graph with adjacency matrix

Find the number of walks of length 2, 3, 4, and 5 from A to D.

01001

10101

01010

0010111010



• Compute the powers of A:

21111

13021

10202

1202011203

24224

42516

25041

2140546152

88668

8153117

6311111

6111938711315

1622141422

2218261034

142642010

1410204262234102618

A2 = A3 =

A4 = A5 =

29



• Thus, for example, there are 34 walks of length 5 from A to D.

• What are they?– A-B-A-B-A-D– A-B-A-E-A-D– A-B-C-B-A-D– A-B-A-D-A-D– etc.


page 651, example 11.3.10

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Trees• a graph with no cycles is called acyclic• forest— acyclic undirected graph, possibly unconnected• dag —- directed acyclic graph• tree (sometimes free tree) —connected forest• A tree is a graph that is connected and circuit-free


1.1 Definition: Let G=(V,E). G is a tree IFF

(a) G is connected; and (b) G does not have any circuits (acyclic).

Comment: • difference between trivial and non-trivial circuits. • A trivial circuit as a circuit of length 0. • All trees are bipartite graphs

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Definition: Let G=(V,E). G is a tree IFF

(a) G is connected; and (b) G does not have any circuits (acyclic).

Example of a tree


Trees

• Theorem: A tree with at least two vertices contains at least one vertex of degree 1.

• Proof:– Begin a simple path at any vertex.– Since G is circuit-free, the path must terminate

at a dead-end.– That dead-end will be a vertex of degree 1.

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• Theorem: A nonempty tree with n vertices contains exactly n – 1 edges.

• Proof (Induction on n):• Basic Step

– Let T be a tree with 1 vertex v.– Any edge must go from v to v, which would be a

circuit.– Therefore, T contains 0 edges.


• Inductive Step– Suppose that, for some n ≥ 1, every tree with n

vertices has exactly n – 1 edges.– Let T be a tree with n + 1 vertices.– Let v be a vertex of degree 1.– Remove v and its incident edge from the graph.– The remaining subgraph is connected and circuit-

free.

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– Therefore, it is a tree with n vertices.– So, it has exactly n – 1 edges.– Since we removed one edge from T, that means

that T has n edges.• Therefore, for all n ≥ 1, a tree with n

vertices has exactly n – 1 edges.


Properties of a Tree

• Theorem: Let G be a graph with n vertices and let p, q, and r be the following statements.– p : G is connected.– q : G is circuit-free.– r : G has exactly n – 1 edges.

Then– p ∧ q → r.– p ∧ r → q.– q ∧ r → p.

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Proof, continued

• Proof: G is connected AND G is circuit-free• Suppose that G is connected and G is

circuit-free.– We have already proven that G must have

exactly n – 1 edges.


Proof, continued

• Suppose that G is connected and G has exactly n – 1 edges.– Suppose G contains a circuit.– Then remove an edge to break the circuit.– Continue to remove edges in this way until the

resulting graph is circuit-free.– Then the resulting graph is connected and

circuit-free, but contains fewer than n – 1 edges.

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Proof, continued

– This is a contradiction.– Therefore, G must be circuit-free.


Proof, continued

• Suppose that G is circuit-free and G has exactly n – 1 edges.– Suppose G is not connected.– Then add an edge connecting two of its

connected components.– Continue adding edges in this manner until the

resulting graph is connected.– None of these edges will create a circuit.

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Proof, concluded

– Therefore, the resulting graph will be connected and circuit-free, but it will contain more than n – 1 edges.

– This is a contradiction.– Therefore, G must have been connected.


Expression Trees

• An expression tree represents an algebraic expression.– Each internal node represents an operator.– Each terminal node represents a number or a

variable.– At each internal node, the left subtree

represents the left operand and the right subtreerepresents the right operand.

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Example: Expression Tree

• The expression (x + 1)/(3x – 4)

1x 4

x3

+

*

–

/


Expression Trees

• If an operator is unary, then– It has a left subtree if it is postfix.– It has a right subtree if it is prefix.

38


Example: Expression Tree

• Represent the expression –(x + 3)4.

4

3x

+

^

–


Binary Search Trees

• A binary search tree is a binary tree with the following logical structure.

• At each node– The values in the left subtree are less than or

equal to the value at the node.– The values in the right subtree are greater than

or equal to the value at the node.

39


• Syntax (or grammatical structure)of natural languages such as English.• Has proved useful in constructing compilers for high level computer

languages.• You define the grammar rules of the language that you’re designing at the

very onset.• These grammar rules ( also called productions) are expressed using a

shorthand notation called Backus-Naur notation.• In BNF, | represents OR and <> are used to enclose terms to be defined.

A very small subset of English grammar can be listed in BNF and applied to a sentence “The young man caught the ball”

1. <sentence> <noun phrase> <verb phrase>2. , 3. < noun phrase > <article> <noun>| <article> <adjective> <noun>4. <verb phrase> <verb> <noun phrase>5. <article> the6. <adjective> young7. <noun> man|ball8. <verb> caught


< sentence >

< noun phrase > < verb phrase >

< article > < adjective> < noun >

<verb ><noun phrase >

caught< article >

< noun >

the ball

the young man

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Let G = (V,E) be an undirected graph. Then the following are equivalent:1. G is a free tree2. any two vertices in G are connected by a unique simple path3. G is connected but if any edge is removed, the resulting graph is disconnectedCompare this to lemma 11.2.1 page 6254. G is connected and | E | = | V | - 15. G is acyclic and | E | = | V | - 16. G is acyclic but if any edge is added, the resulting graph is cyclic

Let T be a tree. A vertex of degree 1 in T is called a terminal vertex(or a leaf), and a Vertex of degree greater than 1 is called an internal vertex(or a branch vertex)

Page 667 example 11.5.4


A rooted tree is a free tree in which one vertex is distinguished from the others.•the distinguished vertex is called the root•vertices in a rooted tree are often called nodes

Let r be the root of a rooted tree T. Note that for any node x, there is a unique path from r to x.•any node y on path from r to x is called an ancestor of x.•If y is an ancestor of x, x is called the descendent of y.

root is only node with no parentsiblings —two nodes which share the same parentexternal node or leaf —node with no childreninternal node – nonleaf nodeNumber of children of a node x in a rooted tree T is called the degree of x.Length of the path from r to x is called the depth of x.largest depth of any node in a tree T is called the height of T

41


level 0

level 1

level 2

level 3

root

2 is ancestor of and the enclosed vertices are all descendents of 2

5 is child of 2 2 is parent of 55, 6, 7 are siblings


Binary treeA binary tree is a rooted tree in which every internal node has atmost two children.Each child in a binary tree is designated either a left child or a right child and an internal vertex has at most one left and one right child. A full binary tree is a binary tree in which each internal vertex has either none or exactly two children.•Left subtree•Right subtree

A

B

D

C

Skinny tree

42


d

/

a +

c

2

+

* ^

473

Binary trees are often used to represent algebraic expressions.Binary trees allow for an "inorder" traversal, where each node is visited after its left child, but before its right child. The preorder, inorder, and postorder traversals of an expression tree will print a prefix, infix, or postfix expression, respectively. •Inorder: visit left subtree, then the root, then the right subtree.•Preorder: visit root, left, right.•Postorder: visit left, right, root.Preorder: + * 3 7 ^ 4 2 Inorder: 3 * 7 + 4 ^ 2 Postorder: 3 7 * 4 2 ^ +

More generally, the binary tree represents a/(c+d).•The internal vertices are arithmetic operators.•Terminal vertices are variables.•Operator at each vertex acts on its left and right subtree in left-right order


Full m-ary Trees (Definition)

• A m-ary tree is FULL iff every vertex has either 0 or m children. (OR every internal vertex has m children).

• Examples of full binary trees.

Full Tree? Yes No Yes No No

Yes Yes

43


Proof:

Full Tree Theorem

• Full Tree theorem: A full m-ary tree with k internal vertices has mk + 1 vertices.

Let T=(V,E) be a full m-ary tree, with k internal vertices.

Total number of vertices in T

Number of vertices that HAVE a parent

Number of vertices that DO NOT HAVE a parent

= +

Q: How many vertices HAVE a parent?

1. Observe for a 2-ary tree with 7 internal vertices

2. Each internal vertex has 2 children


Proof:






= +

Q: How many vertices HAVE a parent?1. Observe for a 2-ary tree with 7

internal vertices2. Each internal vertex has 2 children

Another way of looking at it is that for each internal vertex, there are 2 vertices which have a parent.

44


Proof:






= +





Proof:






= +




45


Proof:






= +





Proof:






= +




46


Proof:






= +





Proof:






= +




47


Proof:






= +

Q: How many vertices HAVE a parent?A: 2k vertices for a binary tree.

mk vertices for a m-ary tree.

(Each internal vertex is the parent of m vertices)

(mk)


Proof:






= +

Q: How many vertices DO NOT HAVE a parent?A: 1. The root does not have a

parent

(mk) (1)

48


Proof:






= +

= mk + 1

(mk) (1)


• Leaves-Height Theorem for Binary Trees: Let T=(V,E) be a binary tree that has t leaves, and height h. Then

t ≤ 2h.

Proof: (by using induction on the height of the tree)

Base Case: h = 0

T has 1 vertex, which is a leaf.

t = 1 ≤ 1 = 20 = 2h

Base case is true.

49


• Leaves-Height Theorem for Binary Trees: Let T=(V,E) be a binary tree that has t leaves, and height h. Then

t ≤ 2h.

Proof: (by using induction on the height of the tree)

Inductive Case: Assume that t ≤ 2h for h = 0,1,2,…,k (STRONG!)

Let T be any binary tree of height k+1. (Need to show t ≤ 2k+1)

k

1

TL TR

With respect to the root vertex, let the left and right subtrees be TL and TR respectively.

Let the number of leaves in TL and TR be tL and tR respectively. (t = tL+ tR)

Height of TL and TR are both < k+1.

By inductive hypothesis, tL ≤ 2(TL Height) ≤ 2k and tR ≤ 2(TR Height) ≤ 2k.

t = tL+ tR ≤ 2k + 2k = 2k+1.


• Corollary to the Leaves-Height Theorem: Let T=(V,E) be a binary tree that has t leaves, and height h. Then

log2 t ≤ h.

Proof:

Using leaves-height theorem, we have t ≤ 2h.

Taking logarithms on both sides will yield

log2t ≤ log22h

log2t ≤ h

50


Examples

• Q: Is there a binary tree that has height 5 and 38 external vertices?

• A: No, since 38 > 25 which violates the leaves-height theorem.


Examples

• Q: Is there a full binary tree with 10 internal vertices and 13 external vertices?

• A: No. Using the full-tree theorem, a binary tree with 10 internal vertices has 21 vertices in total. Therefore there should be 21-10 = 11 external vertices.

51


Leaves-Height Theorem.

IN GENERAL:

• Leaves-Height Theorem for m-ary Trees: Let T=(V,E) be a m-ary tree that has t leaves, and height h. Then

t ≤ mh

• Corollary to the Leaves-Height Theorem: Let T=(V,E) be a m-ary tree that has t leaves, and height h. Then

logmt ≤ h


Spanning Trees

• A spanning tree T of a graph G is a subgraph of G that contains every vertex of G and is a tree.

52


• routing of packets through network switches.• Multiple connections from one switch to the rest of

network to increase fault tolerance.

When all links are operational, redundacy in connection occurs.Network forms a spanning tree so that packets will not be redundantly routed. RNetwork will elect a root.Root will broadcast packets to all other switches.Each switch will select the ‘best’ link to use.


RR

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When a link goes down, the network reconfigures again.


Spanning Trees

• Theorem: Every connected graph has a spanning tree.

• Proof:– If the graph contains circuits, then remove, one

by one, edges that break the circuits until there are no more circuits.

– The resulting subgraph is a tree.

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Weighted Graphs

• A weighted graph G is a triple (V, E, w) where V and E are the vertices and edges and w is a function w : E → R.

• For each edge e, the value w(e) is called the weight of the edge.

• In applications, often the weights represent a cost of some sort.


Examples: Weighted Graphs

• A highway map.– The cities are the vertices.– The highways are the edges.– The distances are the weights.

• A computer network.– The hardware units are the vertices.– The communication lines are the edges.– The cost of installing the lines are the weights.

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Minimal Spanning Trees

• Since the weights often represent a cost of some sort, a goal often is to minimize the total weight of a graph.


Kruskal’s Algorithm

• Let G be a connected weighted graph with nvertices.

• Initialize T to have all the vertices of G, but no edges.

• Let m = 0.

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• While (m < n – 1)– Find an edge e in E of least weight.– Delete e from E.– If the addition of e to the edges of T does not

create a circuit, then• Add e to E(T).• Increment m.



• When the while loop ends, T is a minimal spanning tree of G.

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Example: Kruskal’s Algorithm

• Let the graph be

Add weights

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• Let T be the set of vertices.

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• Add to T the next smallest edge that does not produce a circuit.

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• And so forth.

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Prim’s Algorithm

• Let G be a connected weighted graph.• Choose a vertex v of G and let T be a graph

consisting of only that one vertex.• Let V be all the vertices of G except v.


Prim’s Algorithm

• For i = 1 to n – 1– Find an edge e of G such that

• e connects T to a vertex in V, and• e has the least weight of all edges connecting T to a

vertex in V.

– Add that vertex to T and delete it from V.

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Prim’s Algorithm

• When the for loop ends, T is a minimal spanning tree of G.


Example: Prim’s Algorithm

• Begin with the same graph as in the last example and choose a vertex.

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• Add the edge of least weight incident connecting T to V.

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• Add the remaining edge of least weight incident connecting T to V.

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• And so forth.

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A graph is said to be n-colorable if it's possible to assign one of n colors to each vertex in such a way that no two connected vertices have the same color. The Four Color Theorem asserts that every planar graph - and therefore every "map" on the plane or sphere - no matter how large or complex, is 4-colorable. Despite the seeming simplicity of this proposition, it was only proven in 1976, and then only with the aid of computers. However, because part of the proof consisted of an exhaustive analysis of many discrete cases by a computer, some mathematicians do not accept it. However, no flaws have yet been found, so the proof appears valid.

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Spanning trees algorithms

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A spanning tree for a graph G is a subgraph of G that contains every vertex of G and is a tree.

•Every connected graph has a spanning tree.•Any 2 spanning trees for a graph have the same number of edges.Page 684, example 11.6.1


A weighted graph is a graph for which each edge has an associated real number weight. The sum of the weights of all edges is the total weight of the graph. A minimal spanning tree for a weighted graph is a spanning tree that has that has the least possible total weight compared to all other spanning trees for the graph.

If G is a weighted graph and e is an edge of G then w(e) denotes the weight of E and w(G) denotes the weight of G.

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Kruskal’s and Prim’s theorem.

Page 686, 689

N1 to N2 - cost is 1 - add to tree N7 to N8 - cost is 1 - add to tree N2 to N3 - cost is 2 - add to tree N1 to N6 - cost is 3 - add to tree N2 to N6 - cost is 4 - reject because it forms a circuit N3 to N4 - cost is 4 - add to tree N2 to N7 - cost is 5 - add to tree N3 to N7 - cost is 6 - reject because it forms a circuit N4 to N8 - cost is 6 - reject because it forms a circuit N4 to N7 - cost is 7 - reject because it forms a circuit N4 to N5 - cost is 7 - add to tree

Documents

Example: Graphspot.pcc.edu/~khatwal/Mth_231/Graphtheory.pdf · 1 kavita hatwal math 231 fall 2002 Graph theory a graph is a finite set of dots called vertices (or nodes) connected