How can I prove that by adding one edge to $G$ you create a cycle in $G$?

Question

Any one help me to show the prove for this?

Let the undirected graph $G = (V, E)$ be a tree. Prove that by adding one edge to $G$ you create a cycle in $G$.

Answer 1

Suppose $G$ is a tree — an undirected, acyclic, connected graph.

Suppose $a,b\in G$ and there is no edge from $a$ to $b$. Because $G$ is connected, there is a path from $a$ to $b$ in $G$: $$ a = x_1 \leftrightarrow x_2 \leftrightarrow \dotsc \leftrightarrow x_n = b, $$ where $n > 1$. Then adding an edge $a \leftrightarrow b$ creates a cycle $$ a = x_1 \leftrightarrow x_2 \leftrightarrow \dotsc \leftrightarrow x_n = b \leftrightarrow a, $$ of length $n+1$.