Chapter 8, Problem 1E

For the digraph shown in Fig. 8-25, find

a. the indegree and outdegree of A.

b. the indegree and outdegree of B.

c. the indegree and outdegree of D.

d. the sum of the indegrees of all the vertices.

e. the sum of the outdegrees of all the vertices.

Figure 8-25

To determine

(a)

To find:

The in degree and out degree of A in the given digraph.

Answer to Problem 1E

Solution:

The in degree of A is 3 and out degree of A is 2.

Explanation of Solution

Given:

The given digraph is shown in figure (1).

Figure (1)

Definitions:

Arc:

An arc $XY\left(X\to Y\right)$ indicates that the relationship goes from X to Y.

Indegree:

For a vertex Y, the number of arcs having Y as their ending vertex is called indegree.

Outdegree:

For a vertex X, the number of arcs having X as their starting vertex is called outdegree.

Calculation:

From figure (1) it can be noticed that there are 3 arcs having their ending vertex as A and 2 arcs having their starting vertex as A.

So, the indegree of A is 3 and outdegree of A is 2.

Conclusion:

Thus, the indegree of A is 3 and outdegree of A is 2.

To determine

(b)

To find:

The in degree and out degree of B in the given digraph.

Answer to Problem 1E

Solution:

The in degree of B is 2 and out degree of B is 2.

Explanation of Solution

Given:

The given digraph is shown in figure (2).

Figure (2)

Definitions:

Arc:

An arc $XY\left(X\to Y\right)$ indicates that the relationship goes from X to Y.

Indegree:

For a vertex Y, the number of arcs having Y as their ending vertex is called indegree.

Outdegree:

For a vertex X, the number of arcs having X as their starting vertex is called outdegree.

Calculation:

From figure (2) it can be noticed that there are 2 arcs having their ending vertex as B and 2 arcs having their starting vertex as B.

So, the indegree of B is 2 and outdegree of A is 2.

Conclusion:

Thus, the indegree of B is 2 and outdegree of B is 2.

To determine

(c)

To find:

The in degree and out degree of D in the given digraph.

Answer to Problem 1E

Solution:

The in degree of D is 3 and out degree of D is 0.

Explanation of Solution

Given:

The given digraph is shown in figure (3).

Figure (3)

Definitions:

Arc:

An arc $XY\left(X\to Y\right)$ indicates that the relationship goes from X to Y.

Indegree:

For a vertex Y, the number of arcs having Y as their ending vertex is called indegree.

Outdegree:

For a vertex X, the number of arcs having X as their starting vertex is called outdegree.

Calculation:

From figure (3) it can be noticed that there are 3 arcs having their ending vertex as D and no arc having their starting vertex as D.

So, the indegree of D is 3 and outdegree of D is 0.

Conclusion:

Thus, the indegree of D is 3 and outdegree of D is 0.

To determine

(d)

To find:

The sum of the in degrees of all the vertices.

Answer to Problem 1E

Solution:

The sum of all the indegrees is 10.

Explanation of Solution

Given:

The given digraph is shown in figure (4).

Figure (4)

Definitions:

Arc:

An arc $XY\left(X\to Y\right)$ indicates that the relationship goes from X to Y.

Indegree:

For a vertex Y, the number of arcs having Y as their ending vertex is called indegree.

Outdegree:

For a vertex X, the number of arcs having X as their starting vertex is called outdegree.

Calculation:

From figure (4) it can be noticed that there total 10 arcs, so there will be total 10 indegrees for all the vertices.

Conclusion:

Thus, the sum of all the indegrees is 10.

To determine

(e)

To find:

The sum of the out degrees of all the vertices.

Answer to Problem 1E

Solution:

The sum of all the outdegrees is 10.

Explanation of Solution

Given:

The given digraph is shown in figure (5).

Figure (5)

Definitions:

Arc:

An arc $XY\left(X\to Y\right)$ indicates that the relationship goes from X to Y.

Indegree:

For a vertex Y, the number of arcs having Y as their ending vertex is called indegree.

Outdegree:

For a vertex X, the number of arcs having X as their starting vertex is called outdegree.

Calculation:

From figure (5) it can be noticed that there total 10 arcs, so there will be total 10 outdegrees for all the vertices.

Conclusion:

Thus, the sum of all the outdegrees is 10.