A bistochastic matrix (or doubly stochastic matrix) is a square matrix with real non-negative entries whose row and column sums are equal to $1$. The set of all such matrices, known as the Birkhoff polytope, is the convex-hull of the set of all permutation matrices $P_n$.

In 1959, Marcus and Ree proved that for any bistochastic matrix $E$, $\displaystyle \sum_{i,j=1}^n E_{i,j}^2 \leq \max_{P \in P_n} \mathrm{tr}(P^T E)$. That is, the squared Frobenius norm is bounded above by its maxtrace. Erdös asked for a characterization of those matrices attaining equality. Such matrices are now called Erdös matrices. After remaining largely unexplored for a long time, this class of matrices has received renewed attention leading to interesting structural insights and complete classification up to dimension $4$.

In this talk, I shall provide a structural characterization of Erdös matrices via their zero-entry patterns (or skeletons) after developing the necessary background. This eventually gives a skeleton-based algorithm to enumerate the Erdös matrices and yields a complete classification up to dimension $6$. This talk is based on a joint work with Hariram Krishna, Souvik Pal and G. Krishna Teja.