The Bipartite Zero Forcing Set for a Full Sign Pattern Matrix

Gu-Fang Mou,Zhong-Shan Li,Tian-Fei Wang

doi:10.3390/math8030354

Abstract

For an m × n sign pattern P, we define a signed bipartite graph B ( U , V ) with one set of vertices U = { 1 , 2 , … , m } based on rows of P and the other set of vertices V = { 1 ′ , 2 ′ , … , n ′ } based on columns of P. The zero forcing number is an important graph parameter that has been used to study the minimum rank problem of a matrix. In this paper, we introduce a new variant of zero forcing set−bipartite zero forcing set and provide an algorithm for computing the bipartite zero forcing number. The bipartite zero forcing number provides an upper bound for the maximum nullity of a square full sign pattern P. One advantage of the bipartite zero forcing is that it can be applied to study the minimum rank problem for a non-square full sign pattern.

Highlights

A sign pattern matrix is a matrix whose entries are drawn from {+, −, 0}; a full sign pattern has entries in {+, −}
The signed zero forcing provided an upper bound for the maximum nullity of a given sign pattern, while the signed zero forcing method is unsuitable for studying the minimum rank problem for a non-square sign pattern, such as
We will provide the following Algorithm 1 to construct a sub-signed bipartite graph with the maximum perfect matching M0 corresponding to every set of disjoint M0 -interlacing cycles, which contain an even number of M0 -interlacing e-cycles

Summary

Introduction

A sign pattern matrix is a matrix whose entries are drawn from {+, −, 0}; a full sign pattern has entries in {+, −}. The signed zero forcing provided an upper bound for the maximum nullity of a given sign pattern (see [10]), while the signed zero forcing method is unsuitable for studying the minimum rank problem for a non-square sign pattern, such as. The bipartite zero forcing number provides an upper bound for the maximum nullity of a matrix with a given full sign pattern. One advantage of the bipartite zero forcing set is that it can be applied to study the minimum rank problem for a non-square full sign pattern.

An Algorithm for a Signed Bipartite Graph

Bipartite Zero Forcing Set for a Full Sign Pattern

Conclusions