Finite Connected Simple Graph is Tree iff Size is One Less than Order/Beware

Beware

Just because a simple graph $G$ has an order one greater than its size does not necessarily make it a tree.

$G$ still has to be connected.

Take this simple graph, for instance:

It has order $5$ and size $4$ but is not a tree.

Sources

• 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): $\S 4.1$: The Minimal Connector Problem: An Introduction to Trees: Problem $6$