UGC-NET | UGC NET CS 2014 Dec – II | Question 2

A certain tree has two vertices of degree 4, one vertex of degree 3 and one vertex of degree 2. If the other vertices have degree 1, how many vertices are there in the graph?
(A) 5
(B) n – 3
(C) 20
(D) 11


Answer: (D)

Explanation: There are 2 vertices of degree 4, 1 vcertex of degree 3, 1 vcertex of degree 2 and vertex of degree one is unknown.
Let’s assume k be the nof vertex of degree one.
Total vertex = 2 + 1 + 1 + k = k + 4.
Number of edges = vertex – 1
i.e. k + 4 – 1
= k + 3.
Now apply handshaking lemma(For more information on handshaking lemma Refer:Handshaking Lemma and Interesting Tree Properties)
2 * 4 + 1 * 3 + 1 * 2 + 1 * K = 2 * (No of edges)
i.e. 13 + k = 2 * (k + 3)
k = 7.
Total vertex = 7 + 4 = 11.
So, option (D) is correct.


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