MGF 1107 Section 15.1 Graphs, Paths, and Circuits Objective 1: The ...

MGF 1107
Section 15.1 Graphs, Paths, and Circuits
Objective 1: The learner will use a given graph to answer questions about the graph.
Graph – a finite set of points called vertices, and line segments or curves called edges that start
and end at vertices. Graphs provide a structure for describing relationships. They tell us how a group
of objects are related to one another.
Vertex – a point on the graph.
Edge – Line segment or curve between vertices.
Loop - an edge that starts and ends at the same vertex
Example 1: Name all of the following for the graph below: vertices, edges, loops.
A
D
B
C
Equivalent graphs have the same number of vertices connected to each other in the same way.
Example 2: Explain why the graphs in problem 8 on page 827 are equivalent and draw a third
equivalent graph.
Graphs can provide us with useful information about real – world situations.
Example 3: Work problem 2 on page 827.

Objective 2: The learner will draw a graph to model a specific situation.
Example 4: Work problems 14, 16, and 19 on page 828.
Objective 3: The learner will understand and use the vocabulary of graph theory.
Degree of a vertex – number of edges at that vertex.
Special Case: A loop contributes two to the degree.
Even Vertex – has even degree.
Odd Vertex – has odd degree.
Example 5: Classify each vertex as even or odd.
A B C
D E F

Adjacent Vertices - at least one edge connects the vertices
Path – a sequence of adjacent vertices and the edges connecting them. An edge can only be part of
a path once but vertices can appear more than once in a path.
Example 6: Use vertices to describe three paths that begin at A and end at E .
A B C
D E F
Circuit – a path that begins and ends at the same vertex.
Example 7: Use vertices to describe three circuits that begin (and end) at A .
A B C
D E F
Connected Graph – for any two vertices, at least one path (not necessarily an edge) connects them
Disconnected Graph – a graph that is not connected. A graph that is not connected has separate
pieces (called components) that are connected.
Example 8: Explain why graphs a and b on page 826 are connected but graphs c and d are
disconnected.

graph
Bridge – an edge that if removed from a connected graph would leave behind a disconnected
Example 8: Name the bridge in the graph below.
A B C
D E F
Example 9: Work problems 34 – 48 on page 829.