On the Principal Permanent Rank Characteristic Sequences of Graphs and Digraphs

TL;DR

This paper characterizes the principal permanent rank characteristic sequences of graphs and digraphs, providing constructions and results for various matrix families including nonnegative, symmetric, and skew-symmetric matrices.

Contribution

It offers a complete characterization and construction methods for all principal permanent rank sequences across different matrix classes.

Findings

Characterization of all realizable principal permanent rank sequences.

Construction methods for sequences in nonnegative and symmetric matrices.

Results extending to skew-symmetric matrices.

Abstract

The principal permanent rank characteristic sequence is a binary sequence $r_0 r_1 \ldots r_n$ where $r_k = 1$ if there exists a principal square submatrix of size $k$ with nonzero permanent and $r_k = 0$ otherwise, and $r_0 = 1$ if there is a zero diagonal entry. A characterization is provided for all principal permanent rank sequences obtainable by the family of nonnegative matrices as well as the family of nonnegative symmetric matrices. Constructions for all realizable sequences are provided. Results for skew-symmetric matrices are also included.