Prove that a simple graph is a tree if and only if it is connected but the deletion of any of its edges produces a graph that is not connected.
Can someone here please help me with this proof? Thanks in advance ...
This really depends on the definition of a tree you were given.
But for example if you were given the definition that a tree is a connected graph without any cycles, it follows straight from the definition that the graph is connected, and if you remove a path between a and b, then no path exists between a and b, because otherwise in the original situation there would exist two paths, and thus a cycle.
The same argument works for other definitions like a tree is a connected graph with each vertice connected by only ONE path etc.