GRAPH THEORY and APPLICATIONS PDF

Chapter 10.1 Trees

Trees. Prof. Tesler. Math 184A. Winter 2017. Prof. Tesler. Ch. 10.1: Trees. Math 184A / Winter 2017. 1 / 15. Page 2. Trees. Tree in graph theory. Stick figure

Module 8: Trees and Graphs - Theme 1

Theorem 1. A tree with n nodes has n -1 edges. Proof. Every node except the root has exactly one in-coming edge. Since there

4. Trees

The following result shows the existence of spanning trees in connected graphs. Theorem 4.12 Every connected graph has at least one spanning tree. Proof Let G

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Thus each edge of a directed graph can be drawn as an arrow going from the first vertex to the second vertex of the ordered pair. Graphs: Definitions and Basic

An Introduction to Combinatorics and Graph Theory

available in this pdf file. . w1 . w2 . w3 . w4 . w5 . w6 . w7 . v1 . v2 In general spanning trees are not unique

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1.1s. Write down the number of vertices the number of edges

1.10. Trees and Spanning Trees. 1.10.1. Definition: Tree. ∗ ∗ ∗

3. Page 6. WUCT121. Graphs. 54. When the graph has a large number of vertices it is not easy to find all spanning trees. In fact

An introduction to chordal graphs and clique trees

Clique trees and chordal graphs have carved out a niche for themselves in recent work on sparse matrix algorithms due primarily to research questions

3.1 Characterizations and Properties of Trees 3.2 Rooted Trees

D. Page 7. GRAPH THEORY – LECTURE 4: TREES. 7. Lemma 1.10. Let v and w be two vertices in a tree T such that w is of maximum distance from v (i.e. ecc(v) =

Chapter 10.1 Trees

Trees. Tree in graph theory. Stick figure tree. Not a tree. (has cycle). Not a tree. (not connected). A tree is an undirected connected graph with no cycles

GRAPH THEORY and APPLICATIONS

? Tree: a connected graph with no cycle (acyclic). ? Forest: a graph with no cycle. ? Paths are trees. ? Star: A tree consisting of one vertex adjacent to.

GRAPHS AND TREES

Note that each directed graph has an associated ordinary. (undirected) graph which is obtained by ignoring the directions of the edges. Graphs: Definitions and

3.1 Characterizations and Properties of Trees 3.2 Rooted Trees

GRAPH THEORY – LECTURE 4: TREES. Abstract. Review from §1.5 tree = connected graph with no cycles. Def 1.1. In an undirected tree a leaf is a vertex of ...

An Introduction to Combinatorics and Graph Theory

Graph theory is concerned with various types of networks new tree by introducing a new root vertex and making the children of this root the two.

GRAPH THEORY WITH APPLICATIONS

12.2 The Number of Spanning Trees. Applications. 12.3 Perfect Squares . Appendix I Hints to Starred Exercises. Appendix II Four Graphs and a Table of their

Partitions of Graphs into Trees

Maximal planar bipartite graphs have a 2-tree partition as shown by Ringel [14]. Here we give a different proof of this result with a linear time algorithm.

Chapter 6: Graph Theory

Rather than finding a minimum spanning tree that visits every vertex of a graph an Euler path or circuit can be used to find a way to visit every edge of a

Graphs and Trees

Lots of terminology surrounding graphs tons of types

Graph Theory III - MIT - Massachusetts Institute of Technology

Grow a tree one edge at a time by adding the minimum weight edge of the graph tothe tree making sure that you have a tree at each step ALG2: Select edges one at a time always choosing the minimum weight edge that does notcreate a cycle with previously selected edges

Graph Theory: Trees - IIT Kgp

GRAPH THEORY { LECTURE 4: TREES Abstract x3 1 presents some standard characterizations and properties of trees x3 2 presents several di erent types of trees x3 7 develops a counting method based on a bijection between labeled trees and numeric strings x3 8 showns how binary trees can be counted by the Catalan recursion Outline

Introduction to graph theory - University of Oxford

Trees and forests A tree (a connected acyclic graph) A forest (a graph with tree components) ©Department of Psychology University of Melbourne Bipartite graphs A bipartite graph (vertex set can be partitioned into 2 subsets and there are no edges linking vertices in the same set) A complete bipartite graph (all possible edges are present) K1

Graph Theory I - Properties of Trees

3 Trees Definition 4Given a graph G • A path in G is a sequence of edges such that each edge begins where the previous edge ends and ends where the next edge begins • A cycle in G is a path starting and ending at the same vertex G is called a tree if it contains no cycles

Lecture 12: Introduction to Graphs and Trees

Binary search tree (BST) - a tree where nodes are organized in a sorted order to make it easier to search At every node you are guaranteed: All nodes rooted at the le† child are smaller than the current node value All nodes rooted at the right child are smaller than the current node value

Searches related to trees in graph theory pdf PDF

GRAPH THEORY { LECTURE 5: SPANNING TREES 3 Choosing a Frontier Edge Def 1 3 Let T be a tree subgraph of a graph G and let S be the set of frontier edges for T The function nextEdge(GS) (usually deterministic) chooses and returns as its value the frontier edge in S that is to be added to tree T Def 1 4

What is the difference between a tree and a spanning tree?

Trees and Spanning Trees •A graph having no cycles is acyclic. •A forest is an acyclic graph. •A leaf is a vertex of degree 1. •A spanning sub-graph of G is a sub-graph with vertex set V(G). •A spanning tree is a spanning sub-graph that is a tree.

What is a graph in physics?

Definition 1AgraphGis a setV(G)of points (called vertices) together with a setE(G)of edges connectingthe vertices. Though graphs are abstract objects, they are very naturally represented by diagrams, where we (usually)draw the vertices and edges in the plane.

What is a tree acyclic structure of linked nodes?

Tree- a directed, acyclic structure of linked nodesNode- an object containing a data value and links to other nodes All the blue circles Tree- a directed, acyclic structure of linked nodesEdge- directed link, representing relationships between nodes All the grey lines

How do you implement a graph search algorithm?

Goal: implement the basic graph search algorithms in timeO(m+n). This is linear time, since it takesO(m+n)time simply to read the input. Note that when we work with connected graphs, a running time ofO(m+n)is the same asO(m), sincemn 1. Breadth First Search (BFS)Depth First Search (DFS) Example… Start at the start. Look at all the neighbors.

GRAPH THEORY

Tree: a connected graph with no cycle (acyclic)

Forest: a graph with no cycle

Pathsare trees.

Star: A tree consisting of one vertex adjacent to

Trees are used in many applications:

analysis of algorithms, compilation of algebraic expressions, theoretical models of computation, etc.

Search trees

Abstract models: sort techniques like heapsort.

Proof: by induction on number of edges.

If |E| = 0 then the tree consists of a single isolated vertex. |V| = 1 = |E| + 1 Assume that the theorem is true for trees of at most k edges.

T he edge (y,z) is removed: Two subtrees T 1 and T 2 |V| = |V 1 | + |V 2 | and |E| = |E 1 | + |E 2 | + 1.

Since 0 |E

1 | k and 0 |E 2 | k, |V 1 | = |E 1 | + 1 and |V 2 | = |E 2 | + 1.

Consequently,

|V| = |V 1 | + |V 2 | = |E 1 | + 1 + |E 2 | + 1 = |E 1 | + |E 2 | + 1 + 1 = |E| + 1 y z

for a loop-free undirected graph G = (V, E): A.

G is a tree.

B. G is connected, but the removal of any edge from G disconnects G into two subgraphs that are trees. C.

G contains no cycle, and |V| = |E| + 1.

G is connected, and |V| = |E| + 1.

G contains no cycle, and if a, b ӇV with

A ҧB

If G is a tree, then G is connected.

Let e = (a,b) be any edge of G. Then, ifG-e is connected, there are at least two paths in G from a to b.

From two such paths we can form a cycle.

But G has no cycle.

Hence, G-e is disconnected and the vertices in G-e can be partitioned into two subsets: Vertex a and those vertices that can be reached from a. Vertex b and those vertices that can be reached from b. These two connected components are trees, because a loop or cycle in either component would also be in G. Graph Theory and Applications © 2007 A. Yayimli8Proof (cont.)

B ҧC

If G contains a cycle then let e = (a,b) be an edge of the cycle.

But then, G-e is connected, contradicting the

hypothesis in part B.

So, G contains no cycle.

Since G is a loop-free connected graph, we know that

G is a tree.

Then, |V| = |E| + 1.

C ҧD

Assume G is not connected.

Let G 1 , G 2 , ..., G r be components of G.

For 1 i r, select a vertex v

i ӇG i and add the r-1 edges (v 1 ,v 2 ), (v 2 ,v 3 ), ..., (v r-1 ,v r ) to G to form G'.

G' is a tree.

|V| = |E'| + 1

From C, |V| = |E| + 1, so |E| = |E'| and r-1 = 0

With r = 1, it follows that G is connected.

To complete the proof:

D ҧE ӟEҧA

EccentricityǪ(u) of a vertex u is max

v˝V d(u,v).

The radiusof a graph G is min

u˝V

Ǫ(u)

4 4 4 4

3 3 44

4 3 4 3

Each vertex is labeled

with its eccentricity.

Radius: 2

Diameter: 4 (maximum

Center: The subgraph induced by the vertices of

minimum eccentricity.

Theorem: The center of a tree is a vertex or an

edge (two adjacent vertices).

5 4 4

5 4 3 3 4 5

5 4 4 5

A branch at a nodeu of a tree is a maximal

subtree containing u as an endnode.

The number of branches at u is the degree of u.

The weight at a nodeu is the maximum number of

A node v is a centroid, if v has minimum weight.

The centroid of a treeconsist of all centroid

nodes.

Theorem: The centroid of a tree is a vertex or an

edge (two adjacent vertices). 11

12 6 9 12

Smallest trees with one and two central and

centroid nodes:

1 CENTER 2

In a communication network, large diameter is

acceptable if most pairs can communicate via shortest paths.

We study the average distance

Average: Sum divided by n(n-1)/2. (all pairs)

It is equivalent to study:

This sum is called the Wiener Indexof G.

G uvV

DG d uv

Edges of a tree may be directed.

If is a directed edge, then:

u is the parentof v, v is the childof u. A vertex v is the rootof a directed tree, if there are paths from v to every other vertex in the tree. Graph Theory and Applications © 2007 A. Yayimli17Rooted Tree

Definition: A rooted treeis a tree in which we

identify a vertex v as root (indegree: 0).

Levelof a vertex: Vertices at distance i from the

root lie at level i+1.

The heightof the rooted tree is its maximum

Ordered tree: A directed tree in which the set of

children of each vertex is ordered.

Binary Tree: An ordered tree in which no vertex

has more than two children:

Left child and right child.

Complete binary tree: Every vertex has either

two children or none.

Balancedcomplete binary tree: Every endpoint

A complete balanced binary tree of height h has

2 h - 1 vertices.

A complete balanced N-ary tree of height h has

A cut edgeof G is an edge e such that G - e is

disconnected.This graph has 3 cut edges.

Theorem: A connected graph is a tree if and only

For subsets S and S' of V,

[S,S'] is the set of edges with one end in S, the other in S'.

Edge cut: A subset of E of the form [S,S'], where

S is a nonempty proper subset of V,

S' = V - S.

Bond: A minimal nonempty edge cut of G.

A vertex v is a cut vertex if:

E can be partitioned into two nonempty subsets E

A spanning tree of a connected undirected

graph G is a subgraph which is a tree and which contains all the vertices of G.

The construction of a communication network

A road map or railway system

two nodes, problem is to find a network: at minimum cost and providing route between every two nodes

Solution: The solution is the minimum-weighted

spanning tree of the associated weighted graph.

Minimum-weighted spanning tree can be found

A generalization of the minimum-weighted

spanning tree problem: Given a proper subset V' of the vertices of a graph find a minimum-weighted tree which spans the vertices of V'.

Such a tree is called SteinerTree.

No efficient algorithm is known for Steiner tree

With one or two vertices, only one tree can be

formed.

With three vertices there is one isomorphism

class. The adjacency matrix is determined by which vertex is the center.

So, there are 3 trees with 3 vertices.

With 4 vertices:

There are 4 stars and 12 paths

This yields to 16 trees.

With 5 vertices, a careful study yields 125 trees.

Wit vertices, there are n

The complete graph wit vertices has all the

edges that can be used in forming trees wit vertices.

The number of spanning trees in a complete graph

wit vertices is n n-2

Can we find a method to compute the number of

spanning trees in any graph?

Not containing

the diagonal

Containing

Definition: In a graph G, contractionof edge e

with end points u and v is the replacement of u and v with a single vertex the incident edges of this vertex are the edges other than e that were incident to u and v.

The resulting graph G·e has one less edge

than G. u e v G

G·e

Proposition: Let ˫(G) denote the number of

spanning trees of a graph G. If e ӇE is not a loop, then:

˫(G) = ˫(G-e) + ˫(G·e)

Example:

a b c d e G G - e

4 spanning treesG·e

4 spanning trees

This may lead to a recursive algorithm.

We cannot apply the recurrence when e is a

loop. The loops do not affect the number of spanning trees.

Hence, we can delete loops as they arise.

If we try to compute by deleting and contracting

Form a matrix:

Put the vertex degrees on the diagonal

The remaining elements are 0.

Substract the adjacency matrix from it.

Example:

2000
d0300c0030b0002a d c b a 0110
d1011c1101b0110a d c b a

2-1-10

d -13-1-1 c -1-13-1 b

0-1-12

a d c b a

Matrix for the Kite

Delete a row and a column of the resulting

matrix.

Take the determinant.

Given a loopless graph G:

Vertex set: v

1 , v 2 , ..., v n Let a ij be the number of edges with endpoints v i and v j

Let Q be the matrix in which entry (i,j) is:

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[PDF] GRAPH THEORY and APPLICATIONS

What is the difference between a tree and a spanning tree?

What is a graph in physics?

What is a tree acyclic structure of linked nodes?

How do you implement a graph search algorithm?

GRAPH THEORY

Tree: a connected graph with no cycle (acyclic)

Forest: a graph with no cycle

Pathsare trees.

Star: A tree consisting of one vertex adjacent to

Trees are used in many applications:

Search trees

Abstract models: sort techniques like heapsort.

Proof: by induction on number of edges.

Since 0 |E

Consequently,

G is a tree.

G contains no cycle, and |V| = |E| + 1.

G is connected, and |V| = |E| + 1.

G contains no cycle, and if a, b ӇV with

A ҧB

If G is a tree, then G is connected.

From two such paths we can form a cycle.

But G has no cycle.

B ҧC

But then, G-e is connected, contradicting the

So, G contains no cycle.

G is a tree.

Then, |V| = |E| + 1.

C ҧD

Assume G is not connected.

For 1 i r, select a vertex v

G' is a tree.

From C, |V| = |E| + 1, so |E| = |E'| and r-1 = 0

With r = 1, it follows that G is connected.

To complete the proof:

D ҧE ӟEҧA

EccentricityǪ(u) of a vertex u is max

The radiusof a graph G is min

Ǫ(u)

4 4 4 4

3 3 44

4 3 4 3

Each vertex is labeled

Radius: 2

Diameter: 4 (maximum

Center: The subgraph induced by the vertices of

Theorem: The center of a tree is a vertex or an

5 4 4

5 4 3 3 4 5

5 4 4 5

A branch at a nodeu of a tree is a maximal

The number of branches at u is the degree of u.

The weight at a nodeu is the maximum number of

A node v is a centroid, if v has minimum weight.

The centroid of a treeconsist of all centroid

Theorem: The centroid of a tree is a vertex or an

12 6 9 12

Smallest trees with one and two central and

1 CENTER 2

In a communication network, large diameter is

We study the average distance

Average: Sum divided by n(n-1)/2. (all pairs)

It is equivalent to study:

This sum is called the Wiener Indexof G.

DG d uv

Edges of a tree may be directed.

If is a directed edge, then:

Definition: A rooted treeis a tree in which we

Levelof a vertex: Vertices at distance i from the

The heightof the rooted tree is its maximum

Ordered tree: A directed tree in which the set of

Binary Tree: An ordered tree in which no vertex

Left child and right child.

Complete binary tree: Every vertex has either

Balancedcomplete binary tree: Every endpoint

A complete balanced binary tree of height h has

A complete balanced N-ary tree of height h has

A cut edgeof G is an edge e such that G - e is

Theorem: A connected graph is a tree if and only