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Let G = (V, E) be a graph. Define ξ(G) = ∑ d_i × d_i, where r_i is the number of vertices of degree d_i in G. If S and T are two different trees with ξ(S) = ξ(T), then

1. |S| = 2|T|
2. |S| = |T| - 1
3. |S| = |T|
4. |S| = |T| + 1

Correct answer: |S| = |T|

Solution

The value ξ(G) represents the sum of the squares of the degrees of the vertices in the graph. Since both S and T are trees and have the same number of vertices, their degree distributions must also be equivalent, leading to the conclusion that |S| equals |T|.

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