These seven original questions follow the four lessons in the chapter. Draw the network first, then open the answer.
Questions
Question 1. A network has edges AB, AC, AD and BC. State the degree of each vertex.
Answer
A is on AB, AC and AD, so its degree is 3. B is on AB and BC, so 2. C is on AC and BC, so 2. D is on AD only, so 1.
Check: 3 + 2 + 2 + 1 = 8, and 2 × 4 edges = 8.
Question 2. A network has 5 vertices and 7 edges. Four of the degrees are 4, 3, 3 and 2. Find the fifth degree.
Answer
The sum of degrees is 2 × 7 = 14.
The known degrees add to 4 + 3 + 3 + 2 = 12, so the fifth degree is 14 − 12 = 2.
Question 3. Four players, Hana, Ivan, Jay and Kai, each play a match against some others: Hana v Ivan, Hana v Jay, Ivan v Kai and Jay v Kai. Draw the network, then state the number of edges and each degree.
Answer
Vertices: H, I, J, K. An edge means the two players have played each other.
Edges: HI, HJ, IK and JK, so there are 4 edges.
Each player has played 2 matches, so every degree is 2. The sum is 8, equal to 2 × 4.
Question 4. Four towns are joined by one-way roads, and each road has its length in kilometres written on it. Give two words that describe this graph.
Answer
The one-way roads have arrows, so the graph is directed. The lengths are numbers on the edges, so it is weighted.
Question 5. A road AB has the weight 9 km, and three roads meet at A. A student writes “the degree of A is 9”. What went wrong?
Answer
The 9 is a weight belonging to the edge AB. The degree is a count of edges at a vertex.
Three roads meet at A, so the degree of A is 3.
Question 6. The road lengths in km are AB 6, AC 4, BC 1, BD 3 and CD 7. Find the shortest route from A to D.
Answer
A B D: 6 + 3 = 9. A C D: 4 + 7 = 11.
A B C D: 6 + 1 + 7 = 14. A C B D: 4 + 1 + 3 = 8.
The shortest route is A C B D, with a total of 8 km. Note that the routes with only two roads were not the shortest.
Question 7. Can a simple network have three vertices that each have degree 3? Explain.
Answer
No. The degrees would add to 9, which is odd, but the sum of degrees must equal twice the number of edges, which is always even.
Another reason: in a simple graph with 3 vertices, a vertex can join at most the other 2, so the largest possible degree is 2.
If you got these wrong
Slips in questions 1 and 2 point to identifying vertices, edges and degree. Question 3 needs representing a real network as a graph.
Questions 4, 5 and 7 use distinguishing directed, weighted and simple graphs. Question 6 comes from finding efficient routes in a small weighted network.
Log each slip in the mistake log and paper-error review, then build a timed set with the timed original practice session builder.
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