Complete Bipartite Graphs which are Trees

Theorem
Let $G = \struct {A \mid B, E} =: K_{m, n}$ be the complete bipartite graph with $m$ vertices in $A$ and $n$ vertices in $B$.

Then:
 * $K_{0, 0}$ is a tree
 * $K_{1, n}$ and $K_{n, 1}$ is a tree for all $n$

and no other complete bipartite graphs are trees.

Proof
We note from Null Graph is Complete Bipartite Graph and Null Graph is Tree that $K_{0, 0}$ is a complete bipartite graph that is also a tree.

Next we note that the order of $K_{1, n}$ is $n + 1$.

Indeed, there is $1$ vertex in $A$ and $n$ vertices in $B$, for a total of $n + 1$.

Then we note that the size of $K_{1, n}$ is $n$.

Indeed, each vertex in $B$ has exactly $1$ edge to the $1$ vertex in $A$.

By the definition of complete bipartite graph, that accounts for all edges in $G$.

From Finite Connected Graph is Tree iff Size is One Less than Order it follows that $K_{1, n}$ is a tree.

Similarly:
 * the order of $K_{n, 1}$ is $n + 1$
 * the size of $K_{n, 1}$ is $n$

and again, $K_{n, 1}$ is a tree

Now consider the complete bipartite graph $G = \struct {A \mid B, E} = K_{0, n}$ where $n > 1$.

Let $b_1, b_2 \in B$.

As $K_{0, n}$ is bipartite, $b_1$ and $b_2$ can be adjacent to vertices in $A$ only.

But $A = \O$ and so $b_1$ and $b_2$ are isolated.

Hence $b_1$ and $b_2$ are not connected.

Hence by definition $G$ is not a connected graph

So by definition $K_{0, n}$ is not a tree.

By it follows that $K_{m, 0}$ is also not a tree for $m > 1$.

Now consider the complete bipartite graph $K_{m, n}$ where $m, n > 1$.

We recall that $K_{m, n} := G = \struct {A \mid B, E}$ such that every vertex in $A$ is adjacent to every vertex in $B$.

As $m, n > 1$ we have that:

Hence we can find a cycle in $G$:
 * $\tuple {a_1, b_1, a_2, b_2, a_1}$

As a tree is defined as a (connected) graph with no cycles, it follows that such a $K_{m, n}$ is not a tree.

Hence the result.