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GATE | Gate IT 2007 | Question 25

  • Difficulty Level : Hard
  • Last Updated : 14 Feb, 2018

What is the largest integer m such that every simple connected graph with n vertices and n edges contains at least m different spanning trees?
(A) 1
(B) 2
(C) 3
(D) n

Answer: (C)

Explanation: A graph is connected iff all nodes can be traversed from each node. For a graph with n nodes, there will be n-1 minimum number of edges.
Given that there are n edges, that means a cycle is there in the graph.
The simplex graph with these conditions may be:

Now we can make a different spanning tree by removing one edge from the cycle, one at a time.
Minimum cycle length can be 3, So, there must be atleast 3 spanning trees in any such Graph.

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