GATE CS 2026 Set 2 — Question 36
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Question
Consider a complete graph with vertices (). Note that multiple spanning trees can be constructed over . Each of these spanning trees is represented as a set of edges. The Jaccard coefficient between any two sets is defined as the ratio of the size of the intersection of the two sets to the size of the union of the two sets.Which one of the following options gives the lowest possible value for the Jaccard coefficient between any two spanning trees of ?
Correct answer
(C) 0
Solution
The Jaccard coefficient
J(A, B) between two sets and is defined as:Here, the sets are the sets of edges of spanning trees of . Let and be two spanning trees of . The number of edges in any spanning tree of a graph with vertices is . Thus, .The size of the union is given by .Let be the number of common edges. Then:To find the lowest possible value of the Jaccard coefficient, we need to minimize , the number of common edges.A known result in graph theory states that a complete graph contains edge-disjoint spanning trees. Since , . This means there exist at least two spanning trees and such that they share no edges, i.e., , so .Substituting into the formula:Thus, the lowest possible value is 0.Continue learning with Success Tracker
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