Question:
In a six-node network, two nodes are connected to all the other nodes. Of the remaining four, each is connected to four nodes. What is the total number of links in the network?
In a six-node network, two nodes are connected to all the other nodes. Of the remaining four, each is connected to four nodes. What is the total number of links in the network?
Show Hint
Add up the number of links at every node (the degrees); each link is counted twice in that total.
Updated On: Nov 5, 2026
- 15
- 13
- 7
- 26
Show Solution
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The Correct Option is B
Solution and Explanation
Step 1: Degrees:
Two nodes are connected to all the other 5 nodes, so each has 5 links. The remaining four nodes each have 4 links.
Step 2: Sum of degrees:
\(2 \times 5 + 4 \times 4 = 10 + 16 = 26\).
Step 3: Links:
Every link joins two nodes and is counted twice in the sum, so the number of links is \(26 / 2 = 13\).
Step 4: Check that such a network exists:
The two hubs are joined to each other (1 link) and to each of the other four nodes (8 links). Each of the other four already has 2 links, and needs 2 more among themselves: a 4-cycle gives 4 links. Total \(1 + 8 + 4 = 13\).
Final Answer:
The correct option is (2): 13
Two nodes are connected to all the other 5 nodes, so each has 5 links. The remaining four nodes each have 4 links.
Step 2: Sum of degrees:
\(2 \times 5 + 4 \times 4 = 10 + 16 = 26\).
Step 3: Links:
Every link joins two nodes and is counted twice in the sum, so the number of links is \(26 / 2 = 13\).
Step 4: Check that such a network exists:
The two hubs are joined to each other (1 link) and to each of the other four nodes (8 links). Each of the other four already has 2 links, and needs 2 more among themselves: a 4-cycle gives 4 links. Total \(1 + 8 + 4 = 13\).
Final Answer:
The correct option is (2): 13
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