Question

draw a graph with vertices q, r, s, t, u, v, w, x and with edges connecting the following pairs of vertices: {{q, r}, {q, v}, {r, u}, {r, w}, {r, x}, {s, u}, {s, v}, {t, u}, {v, x}, {w, x}}. for example, {a, b} would represent an edge between vertices a and b.

a. what is the degree of vertex u? 3

b. what is the degree of vertex s? 2

c. how many components does the graph have?

Explanation:

Step1: Analyze the edges connected to each vertex

• For vertex \(U\): The edges \(\{R, U\}\), \(\{S, U\}\), \(\{T, U\}\) are connected to \(U\).
• For vertex \(S\): The edges \(\{S, U\}\), \(\{S, V\}\) are connected to \(S\).
• To find the number of components:
• A component of a graph is a maximal connected sub - graph.
• Let's start building sub - graphs.
• Sub - graph 1:
• Start with \(Q\). Since \(\{Q, R\}\) and \(\{Q, V\}\) are edges. From \(R\), we have edges \(\{R, U\}\), \(\{R, W\}\), \(\{R, X\}\). From \(V\), we have edges \(\{V, X\}\) (already connected through \(R\) - \(X\) and \(V\) - \(X\)) and \(\{S, V\}\). From \(S\), we have \(\{S, U\}\). From \(U\), we have \(\{T, U\}\).
• So, \(Q\), \(R\), \(S\), \(T\), \(U\), \(V\), \(W\), \(X\) are all connected.

Answer:

a. The degree of vertex \(U\) is \(3\).
b. The degree of vertex \(S\) is \(2\).
c. The number of components is \(1\).