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

TL;DR

This paper establishes that any potentially stable 7x7 matrix must have at least 11 non-zero entries, extending known results for smaller matrices and using combinatorial and algebraic methods.

Contribution

It proves a new lower bound on non-zero entries for 7x7 potentially stable matrices and analyzes related digraph properties.

Findings

Potentially stable 7x7 matrices have at least 11 non-zero entries.

All digraphs with up to 10 edges are shown not to correspond to potentially stable matrices.

Minimum edges in strongly connected digraphs depend on their circumference.

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.