Question about regular bipartite graphs A bipartite graph with partite sets U and W is called balanced if |U| = |W|. Let G be a regular bipartite graph with at least one edge. Prove that G is balanced. A regular graph means that every vertex has the same degree this implys that (using the bipartite property) the number of vertices in U = W again by it being bi partite and having vertices of equal degree implys that |U| = |W|. At least in my head that makes sense any hint how to prove it?

Let the degree of every node be $d$ (and since the graph has at least one edge, $d>0$). Then the number of edges associated with nodes in $U$ is $|U|d$, and the number of edges associated with nodes in $W$ is $|W|d$. Every edge in the bipartite graph connects a node in $U$ with a node in $W$, thus the total number of edges associated with the two parts is the same.

Thus $|U|d = |W|d$ and $|U| = |W|$