#### Vol. 3, No. 4, 2010

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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 $i\ne j$) is nonzero whenever $\left\{i,j\right\}$ 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