Aim: Graph Theory – Paths & Circuits Course: Math Literacy Do Now: Aim: What are Circuits and Paths? Can you draw this figure without retracing any of.

Slides:



Advertisements
Similar presentations
The Chinese Postman Problem Route Planning Map Colouring
Advertisements

Chapter 8 Topics in Graph Theory
Section 14.1 Intro to Graph Theory. Beginnings of Graph Theory Euler’s Konigsberg Bridge Problem (18 th c.)  Can one walk through town and cross all.
BY: MIKE BASHAM, Math in Scheduling. The Bridges of Konigsberg.
Euler Circuits and Paths
The Seven Bridges Of Konigsberg.
Pamela Leutwyler. A river flows through the town of Konigsburg. 7 bridges connect the 4 land masses. While taking their Sunday stroll, the people of Konigsburg.
Koenigsberg bridge problem It is the Pregel River divided Koenigsberg into four distinct sections. Seven bridges connected the four portions of Koenigsberg.
Representing Graphs Wade Trappe. Lecture Overview Introduction Some Terminology –Paths Adjacency Matrix.
Euler Graphs Section Euler Graphs 2 Circuit? Path? Non- traversable? A D E C B A D E C B A D E C B End at A End at B Start at A Miss an edge.
Homework collection Thursday 3/29 Read Pages 160 – 174 Page 185: 1, 3, 6, 7, 8, 9, 12 a-f, 15 – 20.
Graphs. Graph A “graph” is a collection of “nodes” that are connected to each other Graph Theory: This novel way of solving problems was invented by a.
MTH118 Sanchita Mal-Sarkar. Routing Problems The fundamental questions: Is there any proper route for the particular problem? If there are many possible.
Graphs and Euler cycles Let Maths take you Further…
Discrete Math Round, Round, Get Around… I Get Around Mathematics of Getting Around.
The Bridge Obsession Problem By Vamshi Krishna Vedam.
Can you find a way to cross every bridge only once?
Chapter 15 Graph Theory © 2008 Pearson Addison-Wesley.
Slide 14-1 Copyright © 2005 Pearson Education, Inc. SEVENTH EDITION and EXPANDED SEVENTH EDITION.
© 2010 Pearson Prentice Hall. All rights reserved. CHAPTER 10 Geometry.
1 Excursions in Modern Mathematics Sixth Edition Peter Tannenbaum.
1 Starter of the day 23 x 27 = x 47 = x 87 = x 55 = x 58 = ???? 54 x 56 = ???? Can you spot the trick for this group of.
5.1  Routing Problems: planning and design of delivery routes.  Euler Circuit Problems: Type of routing problem also known as transversability problem.
Euler Paths & Euler Circuits
Chapter 4 sections 1 and 2.  Fig. 1  Not connected  All vertices are even.  Fig. 2  Connected  All vertices are even.
CSNB143 – Discrete Structure Topic 9 – Graph. Learning Outcomes Student should be able to identify graphs and its components. Students should know how.
Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 5 The Mathematics of Getting Around 5.1Euler Circuit Problems 5.2What.
5.4 Graph Models (part I – simple graphs). Graph is the tool for describing real-life situation. The process of using mathematical concept to solve real-life.
Lecture 14: Graph Theory I Discrete Mathematical Structures: Theory and Applications.
Graph Theory Introducton.
Introduction to Graph Theory
Graphs, Paths & Circuits
Aim: What is an Euler Path and Circuit?
© 2010 Pearson Prentice Hall. All rights reserved. CHAPTER 15 Graph Theory.
Graph Theory Introducton.
Vertex-Edge Graphs Euler Paths Euler Circuits. The Seven Bridges of Konigsberg.
Associated Matrices of Vertex Edge Graphs Euler Paths and Circuits Block Days April 30, May 1 and May
MAT 2720 Discrete Mathematics Section 8.2 Paths and Cycles
SECTION 5.1: Euler Circuit Problems
Chapter 6: Graphs 6.1 Euler Circuits
Chapter 5: The Mathematics of Getting Around
Review Euler Graph Theory: DEFINITION: A NETWORK IS A FIGURE MADE UP OF POINTS (VERTICES) CONNECTED BY NON-INTERSECTING CURVES (ARCS). DEFINITION: A VERTEX.
Chapter 11 - Graph CSNB 143 Discrete Mathematical Structures.
1) Find and label the degree of each vertex in the graph.
Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 14.1 Graphs, Paths, and Circuits.
Copyright © 2009 Pearson Education, Inc. Chapter 14 Section 1 – Slide Graph Theory Graphs, Paths & Circuits.
MAT 110 Workshop Created by Michael Brown, Haden McDonald & Myra Bentley for use by the Center for Academic Support.
Excursions in Modern Mathematics Sixth Edition
Excursions in Modern Mathematics Sixth Edition
CSNB 143 Discrete Mathematical Structures
Çizge Algoritmaları.
Konigsberg’s Seven Bridges
Can you draw this picture without lifting up your pen/pencil?
Euler Paths and Circuits
Euler Circuits and Paths
Euler Paths & Euler Circuits
Graph Theory.
Excursions in Modern Mathematics Sixth Edition
Konigsberg- in days past.
Decision Maths Graphs.
Graph Theory What is a graph?.
Representing Graphs Wade Trappe.
Euler Circuits and Paths
Graphs, Paths & Circuits
Section 14.1 Graphs, Paths, and Circuits
CHAPTER 15 Graph Theory.
Graphs, Paths, and Circuits
Warm Up – 3/19 - Wednesday Give the vertex set. Give the edge set.
Warm Up – 3/14 - Friday 100 seats are to be apportioned.
5 The Mathematics of Getting Around
Presentation transcript:

Aim: Graph Theory – Paths & Circuits Course: Math Literacy Do Now: Aim: What are Circuits and Paths? Can you draw this figure without retracing any of the lines or lifting your pencil off the paper?

Aim: Graph Theory – Paths & Circuits Course: Math Literacy Graphs Eight members of the Soprano family are at a party. The figure below describes unpleasant verbal encounters between members with connecting lines. Graph – provides a structure for describing relationships.

Aim: Graph Theory – Paths & Circuits Course: Math Literacy Graph Theory’s Beginnings In the early 18 th century, the Pregel River in a city called Konigsberg, surround an island before splitting into two. Seven bridges crossed the river and connected four different land area. Many citizens wished to take a stroll that would lead them across each bridge and return them to the starting point without traversing the same bridge twice. Possible?

Aim: Graph Theory – Paths & Circuits Course: Math Literacy Graph Theory’s Beginnings R L A B STYMIED!

Aim: Graph Theory – Paths & Circuits Course: Math Literacy Graph Theory’s Beginnings R L A B FOILED AGAIN! They couldn’t do it. Even you can’t do it! Euler proved that it was not possible.

Aim: Graph Theory – Paths & Circuits Course: Math Literacy not every point where two edges cross is a vertex Definition of Graphs Graph – consists of a finite set of points, called vertices and lines segments or curves, called edges, that start and end at vertices. An edge that starts and ends at the save vertex is called a loop. edge AD edge CC or loop CC

Aim: Graph Theory – Paths & Circuits Course: Math Literacy Graph Equivalence Equivalent – two graphs are equivalent if they have the same number of vertices connected to each other in the same way. Placement of vertices and shapes of edges are unimportant. equivalent both have vertices A, B, C, D both have edges AB, BC, CD same number of vertices connected to each other in the same way makes them equivalent

Aim: Graph Theory – Paths & Circuits Course: Math Literacy Model Problem Are these two graphs are equivalent? equivalent both have vertices A, B, C, D, E both have edges AB, AC, BD, BE, CE, CD, & DE

Aim: Graph Theory – Paths & Circuits Course: Math Literacy Placement of vertices is not related to geographic located. Important is which vertices are connected Same Graph – Different Context

Aim: Graph Theory – Paths & Circuits Course: Math Literacy Graph Theory Definitions Degree of vertex – the number of edges at the vertex Even vertex – even number of edges attached to it Odd vertex – odd number of edges attached to it Adjacent vertices – connected vertices

Aim: Graph Theory – Paths & Circuits Course: Math Literacy Model Problem List the pairs of adjacent vertices. Start with A:A & B, A & E, A & D, A & C B:B & C C:already accounted for D:already accounted for E:already accounted for

Aim: Graph Theory – Paths & Circuits Course: Math Literacy Model Problem The floor plan of a 4-room house is shown. The rooms are labeled A, B, C, and D. The outside is labeled E. The openings represent doors. Draw a graph that models the connecting relationship in the floor plan. Use vertices to represent the rooms and the outside and edges to represent the connecting doors. 2 doors from E to A 2 doors from E to C 1 door from E to D

Aim: Graph Theory – Paths & Circuits Course: Math Literacy Model Problem A mail carrier delivers mail to the four-block neighborhood shown on the next slide. She parks her truck at the intersection shown and then walks to deliver mail to each of the houses. The streets on the outside of the neighborhood have houses on one side only. The interior streets have houses on both sides of the street. On these streets, the mail carrier must walk down the street twice, covering both sides. Draw a graph that models the streets of the neighborhood walked by the mail carrier. Use vertices to represent the street intersections and corners. Use one edge if streets must be covered only once and two edges for streets that must be covered twice.

Aim: Graph Theory – Paths & Circuits Course: Math Literacy Model Problem

Aim: Graph Theory – Paths & Circuits Course: Math Literacy Model Problem A BC F I HG D E

Aim: Graph Theory – Paths & Circuits Course: Math Literacy Model Problem

Aim: Graph Theory – Paths & Circuits Course: Math Literacy Graph Theory Definitions Path – a sequence of adjacent vertices and the edges that connect them. An edge can be part of a path only once.

Aim: Graph Theory – Paths & Circuits Course: Math Literacy Graph Theory Definitions Circuit – a path that begins and ends at the same vertex. Every circuit is a path, but not every path is a circuit.

Aim: Graph Theory – Paths & Circuits Course: Math Literacy Graph Theory Definitions Connected – if for any two of a graph’s vertices there is at least one path connecting them. Disconnected – made up of pieces called components

Aim: Graph Theory – Paths & Circuits Course: Math Literacy The Product Rule