Prove that a simple graph is a tree if and only if it is con

Prove that a simple graph is a tree if and only if it is connected but the deletion of any of its edges products a graph that is not connected.

Solution

Definetion : Tree is a connected simple graph with no cycles (acyclic).

Now, lets assume that our graph is a tree, then it\'ll be connected.

Now if we remove any edge, then the graph is not connected.

Now if we remove an edge between vertices A and B and it\'s still connected.

=> there is a path from v1 to v2 but if we add the removed edge, then we have a cycle.

SO we can see that when the edge is removed and if simple graph has a cycle but since a Tree is acyclic

=> a simple graph is a tree if and only if it\'s connected but the deletion of any of its edges produces a graph that is not connected.