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BY 4.0 license Open Access Published by De Gruyter Open Access December 6, 2020

Maximum nullity and zero forcing of circulant graphs

Linh Duong, Brenda K. Kroschel, Michael Riddell, Kevin N. Vander Meulen and Adam Van Tuyl
From the journal Special Matrices

Abstract

The zero forcing number of a graph has been applied to communication complexity, electrical power grid monitoring, and some inverse eigenvalue problems. It is well-known that the zero forcing number of a graph provides a lower bound on the minimum rank of a graph. In this paper we bound and characterize the zero forcing number of various circulant graphs, including families of bipartite circulants, as well as all cubic circulants. We extend the definition of the Möbius ladder to a type of torus product to obtain bounds on the minimum rank and the maximum nullity on these products. We obtain equality for torus products by employing orthogonal Hankel matrices. In fact, in every circulant graph for which we have determined these numbers, the maximum nullity equals the zero forcing number. It is an open question whether this holds for all circulant graphs.

MSC 2010: 05C50; 05C75; 05C76; 15A03

References

[1] A. Aazami, Hardness results and approximation algorithms for some problems on graphs, PhD thesis, University of Waterloo, 2008. http://hdl.handle.net/10012/4147.Search in Google Scholar

[2] AIM minimum rank - special graphs work group, Zero forcing sets and the minimum rank of graphs. Linear Algebra Appl.428 (2008), 1628–1648.10.1016/j.laa.2007.10.009Search in Google Scholar

[3] J.S. Alameda, E. Curl, A. Grez, L. Hogben, A. Schulte, D. Young, and M. Young, Families of graphs with maximum nullity equal to zero forcing number. Spec. Matrices6:1 (2018), 56–67.10.1515/spma-2018-0006Search in Google Scholar

[4] W. Barrett, H. Van Der Holst, and R. Loewy, Graphs whose minimal rank is two. Electron. J. Linear Algebra11 (2004), 258–280.10.13001/1081-3810.1137Search in Google Scholar

[5] K.F. Benson, D. Ferrero, M. Flagg, V. Furst, L. Hogben, V. Vasilevska, and B. Wissman, Zero forcing and power domination for graph products. Australas. J. Combin.70(2) (2018), 221–235.Search in Google Scholar

[6] F. Boesch, R. Tindell, Circulants and their connectivities. J. Graph Theory8 (1984), 487–499.10.1002/jgt.3190080406Search in Google Scholar

[7] R. Davila, T. Kalinowski, S. Stephen, A lower bound on the zero forcing number. Discrete Appl. Math.250 (2018), 363–367.10.1016/j.dam.2018.04.015Search in Google Scholar

[8] R. Davila, F. Kenter, Bounds for the zero forcing number of graphs with large girth. Theory Appl. Graphs2 (2015), Art. 1, 8 pp.10.20429/tag.2015.020201Search in Google Scholar

[9] G. Davis, G. Domke, 3-Circulant graphs. J. Combin. Math. Combin. Comput.40 (2002), 133–142.Search in Google Scholar

[10] L. Deaett, S. Meyer, The minimum rank problem for circulants. Linear Algebra Appl.491 (2016), 386–418.10.1016/j.laa.2015.10.033Search in Google Scholar

[11] L. DeAlba, J. Grout, L. Hogben, R. Mikkelson, and K. Rasmussen, Universally optimal matrices and field independence of the minimum rank of a graph. Electron. J. Linear Algebra 18 (2009) 403–419.Search in Google Scholar

[12] L. Duong, B.K. Kroschel, Zero Forcing Number (2010). Preprint.Search in Google Scholar

[13] B. Eastman, A LATEX program for creating circulant graphs, GitHub Repository. Available online https://github.com/brydon-zz/circulant, 2014.Search in Google Scholar

[14] S. Fallat, L. Hogben, Minimum rank, maximum nullity, and zero forcing number of graphs. In Handbook of Linear Algebra, 2nd edition, L. Hogben editor, CRC Press, Boca Raton, 2014.10.1201/b16113-53Search in Google Scholar

[15] C. Heuberger, On planarity and colorability of circulant graphs. Discrete Math. 268 (2003), 153–169.10.1016/S0012-365X(02)00685-4Search in Google Scholar

[16] L. Hogben, W. Barrett, J. Grout, H. van der Holst, K. Rasmussen, A. Smith, and D. Young. AIM minimum rank graph catalog, 2016. http://admin.aimath.org/resources/graph-invariants/minimumrankoffamilies/#/cuig.Search in Google Scholar

[17] A. Márquez, A. de Mier, M. Noy, and M.P. Revuelta, Locally grid graphs: classification and Tutte uniqueness. Discrete Math.266 (2003), 327–352.10.1016/S0012-365X(02)00818-XSearch in Google Scholar

[18] S. Meyer, Zero forcing sets and bipartite circulants. Linear Algebra Appl.436 (2012), 888–900.10.1016/j.laa.2011.09.022Search in Google Scholar

[19] M. Muzychuk, Ádám’s conjecture is true in the square-free case. J. Combin. Theory Ser. A72 (1995), 118–134.10.1016/0097-3165(95)90031-4Search in Google Scholar

[20] M. Riddell, The zero forcing number of circulant graphs. MSc Project, McMaster University (2017).Search in Google Scholar

[21] D. West, Introduction to graph theory. Prentice Hall, Inc., Upper Saddle River, NJ, 1996.Search in Google Scholar

Received: 2020-04-23
Accepted: 2020-10-27
Published Online: 2020-12-06

© 2020 Linh Duong et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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