Consider the graph with V = [A, B, C, D, E] and E = [AX, AY, AZ, BB, CX, CY, CZ, YY]. Without drawing a picture of the graph
a. List all the vertices adjacent to Y
b. List all the edges adjacent to
c. Find the degree of Y
d. Find the sum of the degrees of the vertices
a.
A vertex adjacent to Y is one which is linked by an edge (∈E) to Y.
An example from the set E above would be AY. The edge YY is a loop, i.e. it links back to itself, so Y is NOT considered adjacent to Y.
b.
question incomplete.
If the question had been List all the edjacent to X, then it would be all the edges that contain at least one vertex as X:
AX, CX (∈E).
c.
The degree of a vertex is the number of edges that are incident (connected) to the vertex. Loops are counted twice for degree.
d.
Will be left for you as an exercise.
a. The vertices adjacent to Y are: B, C, and Z.
b. The edges adjacent to Y are: AY, YY, and CY.
c. The degree of Y is 3 because it is connected to 3 other vertices: B, C, and Z.
d. To find the sum of the degrees of the vertices, we need to calculate the degree of each vertex and then add them up.
Degree of A = 3 (AX, AY, AZ)
Degree of B = 1 (BB)
Degree of C = 3 (CX, CY, CZ)
Degree of D = 0 (no edges connected to D)
Degree of E = 1 (YY)
Sum of the degrees = Degree of A + Degree of B + Degree of C + Degree of D + Degree of E
= 3 + 1 + 3 + 0 + 1
= 8
To answer these questions, we can use the given vertex set V and edge set E. Without drawing a picture of the graph, we can determine the answers as follows:
a. To list all the vertices adjacent to Y, we need to find all the edges that connect Y to other vertices. From the edge set E, we can see that there are three edges which are adjacent to Y: AY, CY, and YY. Therefore, the vertices adjacent to Y are A, C, and Y itself.
b. To list all the edges adjacent to Y, we already know that AY, CY, and YY are adjacent to Y. Therefore, the edges that are adjacent to Y are AY, CY, and YY.
c. To find the degree of Y, we need to count the number of edges that are adjacent to Y. In this case, we can see that there are three edges adjacent to Y: AY, CY, and YY. Therefore, the degree of Y is 3.
d. To find the sum of the degrees of the vertices, we need to sum up the degrees of all the vertices in the graph. The degree of a vertex is the number of edges that are adjacent to it. From the given vertex set V and edge set E, we can determine the degree of each vertex as follows:
- The degree of A is 1 (AY)
- The degree of B is 1 (BB)
- The degree of C is 2 (CX, CY)
- The degree of D is 0 (no edges adjacent to D)
- The degree of E is 1 (YY)
Now, let's add up the degrees of all the vertices:
Degree(A) + Degree(B) + Degree(C) + Degree(D) + Degree(E) = 1 + 1 + 2 + 0 + 1 = 5
Therefore, the sum of the degrees of the vertices in the graph is 5.