Minimum rank, maximum nullity and zero forcing number for selected graph families

Almodovar, Edgard; DeLoss, Laura; Hogben, Leslie; Hogenson, Kirsten; Murphy, Kaitlyn; Peters, Travis; A. Ramírez, Camila

doi:10.2140/involve.2010.3.371

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-4184 (online)

ISSN 1944-4176 (print)

Author index

To appear

Other MSP journals

Minimum rank, maximum nullity and zero forcing number for selected graph families

Edgard Almodovar, Laura DeLoss, Leslie Hogben, Kirsten Hogenson, Kaitlyn Murphy, Travis Peters and Camila A. Ramírez

Vol. 3 (2010), No. 4, 371–392

DOI: 10.2140/involve.2010.3.371

Abstract

The minimum rank of a simple graph $G$ is defined to be the smallest possible rank over all symmetric real matrices whose $i j$ -th entry (for $i \neq j$ ) is nonzero whenever ${i, j}$ is an edge in $G$ and is zero otherwise. Maximum nullity is taken over the same set of matrices, and the sum of maximum nullity and minimum rank is the order of the graph. The zero forcing number is the minimum size of a zero forcing set of vertices and bounds the maximum nullity from above. This paper defines the graph families ciclos and estrellas and establishes the minimum rank and zero forcing number of several of these families. In particular, these families provide examples showing that the maximum nullity of a graph and its dual may differ, and similarly for the zero forcing number.

Keywords

minimum rank, maximum nullity, zero forcing number, dual, ciclo, estrella

Mathematical Subject Classification 2000

Primary: 05C50, 15A03, 15A18

Milestones

Received: 28 May 2010

Revised: 9 October 2010

Accepted: 10 October 2010

Published: 6 January 2011

Proposed: Chi-Kwong Li

Communicated by Chi-Kwong Li

Authors

Edgard Almodovar
Department of Mathematics University of Puerto Rico, Río Piedras Campus San Juan, PR 00931 United States

Laura DeLoss
Department of Mathematics Iowa State University Ames, IA 50011 United States

Leslie Hogben
Department of Mathematics Iowa State University Ames, IA 50011 United States	American Institute of Mathematics 360 Portage Ave Palo Alto, CA 94306

Kirsten Hogenson
Department of Mathematics University of North Dakota Grand Forks, ND 58202 United States

Kaitlyn Murphy
Montclair State University College of Science and Mathematics Montclair, NJ 07043 United States

Travis Peters
Department of Mathematics Iowa State University Ames, IA 50011 United States

Camila A. Ramírez
Department of Mathematics University of Puerto Rico, Río Piedras Campus San Juan, PR 00931 United States

Vol. 3, No. 4, 2010