Characterization of Erd\"os matrices by their zero entries

TL;DR

This paper characterizes Erd"os matrices, a special class of bistochastic matrices, by their zero entry patterns, provides an algorithm for finding them up to size 6, and identifies some known matrices as Erd"os.

Contribution

It offers a complete characterization of Erd"os matrices based on zero patterns and introduces an algorithm to identify all such matrices up to size 6.

Findings

Each Erd"os matrix pattern has at most one such matrix.

An efficient algorithm finds all Erd"os matrices up to size 6.

Some known RCDS matrices are identified as Erd"os matrices.

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.