Minimum Rank of Graphs Powers Family

Alimohammad M. Nazari, Marzieh Karimi Radpoor

Department of Mathematics, University of Arak, Arak, Iran.
Institute Scientific and Research Bahar, Hamedan, Iran.

DOI: 10.4236/ojdm.2012.22012                

Abstract

In this paper we study the relationship between minimum rank of graph G and the minimum rank of graph for some families of special graph G, where is the jth power of graph G.

Keywords

Minimum Rank; Power of Graphs; Zero Forcing Set

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	AIM Minimum Rank-Special Graphs Work Group, “Zero Forcing Sets and the Minimum Rank of Graphs,” Linear Algebra and Its Applications, Vol. 428, No. 7, 2008, pp. 1628-1648. doi:10.1016/j.laa.2007.10.009
[2]	S. Fallat and L. Hogben, “The Minimum Rank of Symmetric Matrices Described by a Graph: A Survey,” Linear Algebra and Its Applications, Vol. 426, No. 2-3, 2007, pp. 558-582. doi:10.1016/j.laa.2007.05.036
[3]	L. Hogben, “Spectral Graph Theory and the Inverse Eigenvalue Problem of a Graph,” Chamchuri Journal of Mathematics, Vol. 1, No. 1, 2009, pp. 51-72.
[4]	F. Barioli, W. Barrett, S. M. Fallat, H. Tracy Hall, L. Hogben, B. Shader, P. van den Dries Che and Hein van der Holst, “Zero Forcing Parameters and Minimum Rank Problems,” Linear Algebra and Its Applications, Vol. 433, No. 2, 2010, pp. 401-411. doi:10.1016/j.laa.2010.03.008
[5]	L. DeLoss, J. Grout, L. Hogben, T. McKay, J. Smith and G. Tims, “Techniques for Determining the Minimum Rank of a Small Graph,” Linear Algebra and Its Applications, Vol. 432, No. 11, 2010, pp. 2995-3001. doi:10.1016/j.laa.2010.01.008
[6]	F. Barioli, S. M. Fallat and L. Hogben, “A Variant on the Graph Parameters of Colin de Verdi`ere: Implications to the Minimum Rank of Graphs,” Electronic Journal of Linear Algebra, Vol. 13, 2005, pp. 387-404.
[7]	L. Hogben and H. van der Holst, “Forbidden Minors for the Class of Graphs G with ,” Linear Algebra Applications, Vol. 423, No. 1, 2007, pp. 42-52. doi:10.1016/j.laa.2006.08.003
[8]	A. Berman, S. Friedland, L. Hogben, U. G. Rothblum and B. Shader, “An Upper Bound for the Minimum Rank of a Graph,” Linear Algebra and Its Applications, Vol. 429, No. 7, 2008, pp. 1629-1638. doi:10.1016/j.laa.2008.04.038
[9]	L. DeLoss, J. Grout, L. Hogben, T. McKay, J. Smith and G. Tims, “Table of Minimum Ranks of Graphs of Order at Most 7 and Selected Optimal Matrices,” http://arxiv.org/abs/0812.0870
[10]	American Institute of Mathematics Workshop, “Spectra of Families of Matrices Described by Graphs, Digraphs, and Sign Patterns,” 23-27 October 2006, Palo Alto.