Vol. 3, No. 4, 2010

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ISSN: 1944-4184 (e-only)
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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
Abstract

The minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ij-th entry (for ij) 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