Honors Discrete 7.1 – 7.3 Worksheet:
USE A SEPARATE SHEET OF PAPER AS NEEDED!!!!
1) Determine if the following descriptions will (1) ALWAYS, (2) NEVER, or
(3) SOMETIMES be a tree. If the graph is sometimes a tree, then make sure to
provide an example of the graph as a tree and another not as a tree.
a. 5 vertices and exactly one path
from any vertex to any other vertex.
f. 5 bridges and 6 vertices.
g. 3 vertices and no circuits
b. 27 edges and 26 vertices.
h. Euler Path Exists
c. 5 vertices and 4 edges.
d. 15 vertices and 14 bridges.
e. 4 vertices and no circuits.
i. Connected graph with
Redundancy = 0
j. G is connected with 6 vertices and
every vertex has degree 5.
2) Which of the following graphs is a tree?
 If it is not a tree, create a spanning tree.
 Hint: Calculate the redundancy and check if it is a network.
GRAPH #1
GRAPH #4
GRAPH #2
GRAPH #5
GRAPH #3
GRAPH #6
3) Draw a tree with each of the following requirements.
a. 5 vertices and one vertex of
c. 8 vertices and one vertex of
degree 2.
degree 4 and one of degree 3.
b. 6 vertices and two vertices of
degree 3.
d. 7 vertices with two vertices of
degree 2.
4) Find the total number of spanning trees in each of the below graphs.
5) Find the minimum spanning tree in each of the graphs below: Highlight,
Trace, or identify the edges and the total weight.
12
8
11
15
7
12
17
10
6
11
23
8
16
15
12
19
8
5
9
13
Honors Discrete 7.1 – 7.3 Worksheet: SOLUTIONS
1) Determine if the following descriptions will (1) ALWAYS, (2) NEVER, or
(3) SOMETIMES be a tree. If the graph is sometimes a tree, then make sure to
provide an example of the graph as a tree and another not as a tree.
a. 5 vertices and exactly one path
from any vertex to any other vertex.
f. 5 bridges and 6 vertices.
ALWAYS
ALWAYS
g. 3 vertices and no circuits
b. 27 edges and 26 vertices.
SOMETIMES
NEVER
h. Euler Path Exists
c. 5 vertices and 4 edges.
SOMETIMES
SOMETIMES
d. 15 vertices and 14 bridges.
i. Connected graph with
Redundancy = 0
ALWAYS
e. 4 vertices and no circuits.
SOMETIMES
ALWAYS
j. G is connected with 6 vertices and
every vertex has degree 5.
NEVER
2) Which of the following graphs is a tree?
 If it is not a tree, create a spanning tree.
 Hint: Calculate the redundancy and check if it is a network.
GRAPH #1 = YES
R=0
GRAPH #2 = YES
R=0
GRAPH #3 = NO
R=0
#2 continued)
GRAPH #4
NO: R = 1
GRAPH #5
YES: R = 0
GRAPH #6
YES: R = 0
3) Draw a tree with each of the following requirements.
e. 5 vertices and one vertex of
g. 8 vertices and one vertex of
degree 2.
degree 4 and one of degree 3.
f. 6 vertices and two vertices of
degree 3.
h. 7 vertices with two vertices of
degree 2.
4) Find the total number of spanning trees in each of the below graphs.
5*4 – 1 =
19 Spanning Trees
6*4*3 =
72 Spanning Trees
5 Spanning Trees
(4*4 – 1) (3*4 – 1) =
165 Spanning Trees
5*(3*3 – 1) =
40 Spanning Trees
5) Find the minimum spanning tree in each of the graphs below: Highlight,
Trace, or identify the edges and the total weight.
12
8
11
15
7
12
17
10
6
11
23
8
16
15
12
8
5
13
19
Total Weight =
50
9
Total Weight =
108