Graph Theory

Introducton

Graph Theory

Vertex: A point. An intersection of two lines (edges).

Edge: A line (or curve) connecting two vertices.

Loop: An edge that connects a vertex to itself only.

Graph TheoryEx) Represent the "Konigsberg Bridge“

problem using a vertex-edge graph.

* Vertices represent locations.* Edges represent “connections” between those locations.

A

B C

D

A

B C

D

Graph TheoryEx) Represent this map using a vertex-edge graph.

Hint: On map problems, place vertices relative to their actual locations on the map.

O

K

CU

W

N* Edges represent borders in a map problem.

Graph TheoryEx) Represent a floor plan using a vertex-edge graph.

Outside P

HF

J

C

M

O

Graph TheoryThe degree of a vertex is the number of edges "entering" the vertex.

Degree2 1

2

Degree3

12 3 Degree

41

2 3

4

Graph TheoryOdd and Even vertices

If the degree of a vertex is an odd number, then the vertex is considered an odd vertex.

If the degree of the vertex is an even number, then the vertex is considered an even vertex.

Graph TheoryEx) How many odd vertices are there in this graph?

Degree4

Degree4

Degree3

Degree4

Degree1

2 odd vertices F

Degree0

Graph TheoryA path is a sequence of adjacent vertices and the edges connecting them.

Given the graph to the left, some examples of paths could be:

ABC

BCDD

CDBCA

Graph TheoryA circuit is a path that begins and ends at the same vertex.

Given the graph to the left, some examples of circuits could be:

ABCACDDBAC

Graph TheoryOn a connected graph, you can draw a path from one vertex to any other vertex.

Graph TheoryIf a graph is not connected, it is disconnected.

Graph TheoryA bridge is an edge that if removed from a connected graph would create a disconnected graph.

Graph Theory