In Exercises 1-6, do the following: | |
a. | Determine the number of vertices edges, and loops in the given graph |
b. | Draw another representation of the graph that looks significantly different. |
c. | Find two adjacent edges. |
d. | Find two adjacent vertices |
e. | Find the degree of each vertex. |
f. | Determine whether the graph is connected. |
a)
The number of vertices, edges, and loops in the given graph.
Answer to Problem 1CR
Solution:
The number of vertices are
Explanation of Solution
Given:
The following figure shows the given diagram.
Figure
Approach:
A graph is the diagram that consists of points, known as vertices, and connecting lines are known as edges. The edge that connects a vertex with itself is called a loop.
Therefore, the number of vertices are
Conclusion:
Hence, the number of vertices are
b)
To draw:
The representation of the graph that looks significantly different.
Answer to Problem 1CR
Solution:
The representation of the graph that looks significantly different is shown in figure
Explanation of Solution
Approach:
The significantly different representation is as follows,
Take three vertices as A, B and C. Connect A and B with two edges, A and C with two edges. Make an edge between B and C. To form a loop make an edge that connects the vertex B with each other as shown below.
Figure
Therefore, the representation of the graph that looks significantly different is shown in figure
Conclusion:
Hence, the representation of the graph that looks significantly different is shown in figure
c)
To find:
The two adjacent edges of the given figure.
Answer to Problem 1CR
Solution:
The adjacent sides are AB, BC.
Explanation of Solution
Approach:
Adjacent edges are the edges which have a common vertex.
In the above given figure the adjacent sides are AB, BC.
Therefore, the adjacent sides are AB, BC.
Conclusion:
Hence, the adjacent sides are AB, BC.
d)
To find:
The two adjacent vertices.
Answer to Problem 1CR
Solution:
The vertices A and B, B and C are two adjacent vertices.
Explanation of Solution
Approach:
Two vertices will be adjacent if they are joined by an edge.
Thus in the above given figure Vertices A and B, B and C are two adjacent vertices.
Therefore, the vertices A and B, B and C are two adjacent vertices.
Conclusion:
Hence, the vertices A and B, B and C are two adjacent vertices.
e)
To find:
The degree of each vertex.
Answer to Problem 1CR
Solution:
The degree of each vertex is shown in the table (1).
Explanation of Solution
Approach:
The degree of vertex is defined as the total number of edges that are connected to it.
The degree of vertex is shown in the following table.
Vertex | Degree |
A | |
B | |
C |
Table (1)
Therefore, the degree of each vertex is shown in the table (1).
Conclusion:
Hence, the degree of each vertex is shown in the table (1).
f)
Whether the graph is connected or not.
Answer to Problem 1CR
Solution:
The given graph is said to be connected.
Explanation of Solution
Approach:
A graph will be connected if every pair of vertices in the graph is connected by a trail. A trail is sequence of adjacent vertices and the distinct edges that connect to them. Consider the above graph. There is a trail of
Therefore, the given graph is said to be connected.
Conclusion:
Hence, the given graph is said to be connected.
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