Consider the graph with V = [A, B, C, X, Y, Z] 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 AY.
c. Find the degree of Y.
d. Find the sum of the degrees of the vertices.
V is a set of vertices.
E is a set of edges (lines joining the vertices)
(a)
Two distinct vertices are adjacent when there is an edge joining the two.
So to list vertices adjacent to Y , we need to go through all the edges and select those that include Y as a vertex. The other vertex associated with the same edge is therefore adjacent to Y.
There should be two.
(b) Similarly, edges adjacent to AY are those that include either A or Y in the adjacency list (set E).
(c) degree of a vertex is the number of edges that are incident to it.
(d) add the degrees of all the vertices.
To answer the questions, we need to understand the concept of adjacency and vertex degrees in a graph.
a. To list all the vertices adjacent to Y, we need to identify all the vertices that share an edge with Y. In this case, Y is adjacent to A and Z. Therefore, the vertices adjacent to Y are A and Z.
b. To list all the edges adjacent to AY, we need to identify all the edges that are connected to the vertex Y. In this case, the only edge adjacent to Y is AY itself.
c. The degree of a vertex is the number of edges that are connected to that vertex. To find the degree of Y in this graph, we count the number of edges that are connected to Y. In this case, Y is connected to the edges AY, YY, and CY. Therefore, the degree of Y is 3.
d. The degree of a vertex is the sum of the degrees of all vertices in the graph. To find the sum of the degrees of the vertices, we need to calculate the degree of each vertex and add them up.
Degree of A: 3 (AX, AY, AZ)
Degree of B: 2 (BB)
Degree of C: 3 (CX, CY, CZ)
Degree of X: 0
Degree of Y: 3 (AY, YY, CY)
Degree of Z: 1 (AZ)
Sum of degrees = Degree of A + Degree of B + Degree of C + Degree of X + Degree of Y + Degree of Z
= 3 + 2 + 3 + 0 + 3 + 1
= 12
Therefore, the sum of the degrees of the vertices in this graph is 12.