1. 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?

2. 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.

3. 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?

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

5. 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.

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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.

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

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

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

17. 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.

18. 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.

19. 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

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