Chapter 9.CR, Problem 1CR

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.

To determine

a)

The number of vertices, edges, and loops in the given graph.

Answer to Problem 1CR

Solution:

The number of vertices are $3$, number of edges are $6$, and number of loops is zero.

Explanation of Solution

Given:

The following figure shows the given diagram.

Figure $\left(1\right)$

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 $3$, number of edges are $6$, and number of loops is zero.

Conclusion:

Hence, the number of vertices are $3$, number of edges are $6$, and number of loops is zero.

To determine

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 $\left(2\right)$.

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 $\left(2\right)$

Therefore, the representation of the graph that looks significantly different is shown in figure $\left(2\right)$.

Conclusion:

Hence, the representation of the graph that looks significantly different is shown in figure $\left(2\right)$.

To determine

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.

To determine

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.

To determine

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	$4$
B	$6$
C	$2$

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).

To determine

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 $\text{A}\to \text{B}\to \text{C}$.

Therefore, the given graph is said to be connected.

Conclusion:

Hence, the given graph is said to be connected.