Let G=(V,E) be a connected graph where V is the set of vertices and E is the set of edges. The graph G has a property such that for any two vertices v_i,v_j∈V, there is a unique simple path between them. Prove that G is a tree and find the number of spanning trees in G if ∣V∣=n. Additionally, if G is labeled, meaning each vertex v_i has a unique label from 1 to n, determine the number of labeled spanning trees that can be formed from G.

Let G=(V,E) be a connected graph where V is the set of vertices and E is the set of edges. The graph G has a property such that for any two vertices v_i,v_j∈V, there is a unique simple path between them. Prove that G is a tree and find the number of spanning trees in G if ∣V∣=n. Additionally, if G is labeled, meaning each vertex v_i has a unique label from 1 to n, determine the number of labeled spanning trees that can be formed from G.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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