Graph TheoryGraph Theory

Emily BelyeaPaton VinalPaul Friedman

Emily BelyeaPaton VinalPaul Friedman

DefinitionsDefinitions A graph consists of two types of elements, vertices and

edges. Every edge has two endpoints in the set of vertices, and connects or joins the two endpoints.

A cycle is edges connected that make a continuous circuit.

A graph consists of two types of elements, vertices and edges. Every edge has two endpoints in the set of vertices, and connects or joins the two endpoints.

A cycle is edges connected that make a continuous circuit.

Vertex

Edge

A vertex is simply drawn as point or dot. The vertex set of G is usually represented by V(G), or V when there is no danger of confusion. The order of a graph is the number of its vertices, for example, |V(G)|.

An edge is drawn as a line connecting two vertices, called end vertices. An edge with end vertices x and y is represented by xy. The edge set of G is usually written as E(G), or E when there is no possible confusion.

cycle

A bridge, or cut edge or isthmus, is an edge whose removal disconnects a graph, meaning there can only be one line connecting two vertices.

A tree is a connected graph that contains no cycles.

A bridge, or cut edge or isthmus, is an edge whose removal disconnects a graph, meaning there can only be one line connecting two vertices.

A tree is a connected graph that contains no cycles.

Bridge

tree

A weighted graph associates a label, a weight, with every edge found in the graph. Weights are usually listed as real numbers. The numbers are usually restricted to rational numbers or integers. Certain algorithms require further restrictions on weights.

Acyclic graph is a graph that contains no cycles.

A weighted graph associates a label, a weight, with every edge found in the graph. Weights are usually listed as real numbers. The numbers are usually restricted to rational numbers or integers. Certain algorithms require further restrictions on weights.

Acyclic graph is a graph that contains no cycles.

3

54

4

7

567

Weighted acyclic graph

The ProblemThe Problem

Our farm is installing a new irrigation system for a field. The well is located off the field and is the water source for the system. There is a need to find the amount of piping that will be used for the system. Each vertex represents a sprinkler head that provides water to the field. What is the minimum amount of piping that can be used for the field?

Our farm is installing a new irrigation system for a field. The well is located off the field and is the water source for the system. There is a need to find the amount of piping that will be used for the system. Each vertex represents a sprinkler head that provides water to the field. What is the minimum amount of piping that can be used for the field?

The FieldThe Field

WELL

17

15

18

16201921

23

24

14 22 25

17

26

27

30

23

28

29 27

161918

24

Total weight of the TreeTotal weight of the Tree

In order to find the total weight of the tree, add up the total of the weighted edges.

14+15+16+16+17+17+18+18+19+19+20+21+22+23+23+24+24+25+26+27+27+28+ 29+30=518

In order to find the total weight of the tree, add up the total of the weighted edges.

14+15+16+16+17+17+18+18+19+19+20+21+22+23+23+24+24+25+26+27+27+28+ 29+30=518

Minimal Spanning Tree Algorithm

Minimal Spanning Tree Algorithm

1. Each edge of the graph needs to be labeled, starting with the smallest amount weighted to the largest. (e1,e2,e3…en) weighted as:the weight of e1<e2<en

2. Start the graph sequence with e1.3. Continue with the next smallest

weighted edge, and continue until there is none left without making a circuit.

1. Each edge of the graph needs to be labeled, starting with the smallest amount weighted to the largest. (e1,e2,e3…en) weighted as:the weight of e1<e2<en

2. Start the graph sequence with e1.3. Continue with the next smallest

weighted edge, and continue until there is none left without making a circuit.

The Field (labeled weighted)

The Field (labeled weighted)

WELL

e5

e2

e8

e4e11e10e12

e14

e17

e1 e13 e18

e6

e19

e20

e24

e15

e22

e23 e21

e3e9e7

e16

The Field (removing unnecessary pipes)The Field (removing unnecessary pipes)

WELL

e5

e2

e8

e4e11e10e12

e14

e17

e1 e13 e18

e6

e15

e22

e21

e3e9e7

e16

The dotted lines are the lines that are unnecessary for the irrigation system, they are removed.

The Field as a Minimal Spanning Tree

The Field as a Minimal Spanning Tree

WELL

e5

e2

e8

e4e11e10e12

e14

e17

e1 e13 e18

e6

e15

e22

e21

e3e9e7

e16

Totaling the Minimal Spanning Tree

Totaling the Minimal Spanning Tree

14+15+16+16+17+17+18+18+19+19+20+21+22+23+23+24+24+25+27+28=406

The new total saves the farm 112 feet of piping, saving 21.62%

14+15+16+16+17+17+18+18+19+19+20+21+22+23+23+24+24+25+27+28=406

The new total saves the farm 112 feet of piping, saving 21.62%

THE ENDTHE END