Characterization of Erd\"os matrices by their zero entries

Abstract

An Erd\"os matrix $E$ is a bistochastic matrix whose sum of squares of entries (Frobenius norm squared) equals its maxtrace (maximum of all the $\sigma$-traces for permutations $\sigma$'s). We characterize all Erd\"os $E$ by the patterns of their zero entries; showing that each such skeleton has at most one $E$. We present an algorithm to find all $n\times n$ Erd\"os matrices, which finds them up to $n\leqslant 5$ quickly and also size $n=6$. We further show some presently known RCDS matrices to be Erd\"os.