Minimum number of non-zero-entries in a $7\times 7$ stable matrix

Abstract

We prove that if a $7\times 7$ matrix is potentially stable, then it has at least 11 non-zero entries. The results for $n\times n$ matrix with $n$ up to 6 are known previously. We prove the result by making a list of possible associated digraphs with at most 10 edges, and then use algebraic conditions to show all of these digraphs or matrices cannot be potentially stable. In relation to this, we also determine the minimum number of edges in a strongly connected digraph depending on its circumference.