NUMBERS OF NODES AND LINKS, CONNECTIVITY, AND LOOPS

Given a graph with N nodes.
 (a) If it has less than N-1 links then it cannot be connected.
 (b) If it has more than N-1 links and is connected then it must have a cycle.
 (c) If it has exactly N-1 links, it may or may not be a tree.
       but if in addition it is connected then it must be a tree.

A graph is a tree if and only if it is connected and the number of nodes is exactly one more than the number of links.