True or false: If an undirected graph 𝐺G on 𝑛n vertices is connected and has at most 𝑛−1n−1 edges then 𝐺G must be a tree.TrueFalseSubmit

Answer: True

To determine if the statement is true or false, let's analyze the definition and properties of trees and graphs.

1. Definition of a Tree: A tree is defined as a connected acyclic graph. For a graph with $n$ vertices to be a tree, it must have exactly $n - 1$ edges.

2. Properties of Connected Graphs:

   • A connected graph is one in which there is a path between every pair of vertices.
   • A connected graph with $n$ vertices must have at least $n - 1$ edges to ensure connectivity.

3. Examining the Claims:

   • If the undirected graph $G$ is connected and has at most $n - 1$ edges, it cannot contain cycles because adding even a single edge beyond $n - 1$ edges would necessarily create a cycle or disconnect the graph.
   • Given the condition that $G$ has at most $n - 1$ edges and is connected, it follows that $G$ must indeed have $n - 1$ edges to satisfy the conditions of being a tree (connected and acyclic).

Therefore, if an undirected graph $G$ on $n$ vertices is connected and has at most $n - 1$ edges, then it must be a tree.

Final Answer

True