A tree has \(2𝑛\) vertices of degree 1,\( 3𝑛\) vertices of degree 2, \(𝑛\)…
2019
A tree has \(2𝑛\) vertices of degree 1,\( 3𝑛\) vertices of degree 2, \(𝑛\) vertices of degree 3. Determine the number of vertices and edges in tree.
Answer: A. 12,11 — Step 1: Count vertices. Total vertices V = 2n + 3n + n = 6n. Step 2: Sum of degrees and edges. Sum of degrees = 1·(2n) + 2·(3n) + 3·n = 11n. By the handshake…
- A.
12,11
- B.
11,12
- C.
10,11
- D.
9,10
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Correct answer: A
Step 1: Count vertices.
Total vertices V = 2n + 3n + n = 6n.
Step 2: Sum of degrees and edges.
Sum of degrees = 1·(2n) + 2·(3n) + 3·n = 11n. By the handshake lemma, 2E = 11n, so E = 11n/2.
Step 3: Use the tree property.
For a tree, E = V − 1, so E = 6n − 1.
Equate the two expressions for E: 11n/2 = 6n − 1.
Solve: multiply both sides by 2 → 11n = 12n − 2 ⇒ n = 2.
Conclusion: With n = 2, V = 6n = 12 and E = V − 1 = 11.
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