A tree has \(2π\) vertices of degree 1,\( 3π\) vertices of degree 2, \(π\)β¦
2019
A tree hasΒ \(2π\)Β vertices of degreeΒ 1,\(Β 3π\)Β vertices of degreeΒ 2,Β \(π\)Β vertices of degreeΒ 3. Determine the number of vertices and edges in tree.
- A.
12,11
- B.
11,12
- C.
10,11
- D.
9,10
Attempted by 106 students.
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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 = 1Β·(2n) + 2Β·(3n) + 3Β·n = 11n. By the handshake lemma, 2E = 11n, so E = 11n/2.
Step 3: Use the tree property.
For a tree, E = V β 1, so E = 6n β 1.
Equate the two expressions for E: 11n/2 = 6n β 1.
Solve: multiply both sides by 2 β 11n = 12n β 2 β n = 2.
Conclusion: With n = 2, V = 6n = 12 and E = V β 1 = 11.
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