Patterns of Alternating Sign Matrices

TL;DR

This paper explores the zero-nonzero patterns of alternating sign matrices, characterizing their sum vectors, minimum nonzero entries, and properties of symmetric and maximal patterns, linking matrix patterns to graph theory.

Contribution

It introduces a comprehensive analysis of the patterns of alternating sign matrices, including characterizations and bounds for various classes, and connects these patterns to graph-theoretic concepts.

Findings

Characterized row and column sum vectors of alternating sign matrices.

Determined the minimum number of nonzero entries in connected matrices.

Identified maximum edges in symmetric alternating sign matrices without loops.

Abstract

We initiate a study of the zero-nonzero patterns of n by n alternating sign matrices. We characterize the row (column) sum vectors of these patterns and determine their minimum term rank. In the case of connected alternating sign matrices, we find the minimum number of nonzero entries and characterize the case of equality. We also study symmetric alternating sign matrices, in particular, those with only zeros on the main diagonal. These give rise to alternating signed graphs without loops, and we determine the maximum number of edges in such graphs. We also consider n by n alternating sign matrices whose patterns are maximal within the class of all n by n alternating sign matrices.