We characterize the bipartite graphs that minimize the (first degree-based) entropy among all bipartite graphs of given size. For bipartite graphs given size and (upper bound on the) order, we give a lower bound for this entropy. The extremal graphs turn out to be complete bipartite graphs, or nearly complete bipartite. Here we make use of an equivalent representation of bipartite graphs by means of Young diagrams, which make it easier to compare the entropy of related graphs. We conclude that the general characterization of the extremal graphs is a difficult problem, due to its connections with number theory. However, it is easier to identify them for particular values of the order n and size m because we have narrowed down the possible extremal graphs. We indicate that some of our ideas extend to other degree-based topological indices as well. •Characterize the bipartite graphs that minimize the first-degree based entropy given the size or both the size and the order.•Prove that the extremal graphs turn out to be complete bipartite graphs, or nearly complete bipartite.•Make use of an equivalent representation of bipartite graphs by means of Young diagrams.•Extend our ideas to some degree-based topological indices.