ZERO FORCING NUMBER AND MAXIMUM NULLITY OF GENERAL POWER GRAPHS

Document Type : Original Manuscript

Authors

1 Department of Mathematics, Faculty of Basic Science, Imam Khomeini International University, P.O. Box 34148-96818, Qazvin, Iran

2 Department of Mathematics, Faculty of Basic Science, Imam Khomeini International University, P.O. Box 34148-96818, Qazvin, Iran.

Abstract

Let Γ = (V,E) be a simple and undirected graph. General power graph of Γ, shown by Pg(Γ), is a graph with the vertex set P(V (Γ))\ϕ. Also two distinct vertices of B and C are adjacent if and only if every b ∈ B is adjacent to every c ∈ C \{b} in Γ. In this paper, we consider general power graph related to graph Γ. Also we show that zero forcing number is equal to maximum nullity, for general power graph of some graphs.

Keywords

References

1. E. Almodovar, L. DeLoss, L. Hogben, K. Hogenson, K. Murphy, T. Peters and C. A. Ramírez, Minimum rank, maximum nullity and zero forcing number for selected graph families, Involve, a Journal of Mathematics, 3(4) (2010), 371–392.
 
2. A. R. Ashrafi , A. Hamzeh and S. Hossein Zadeh, Calculation of some topological indices of splices and links of graphs, J. Appl. Math. Inform., 29 (2011), 327–335.
 
3. F. Barioli, W. Barrett, S. Butler, S. M. Cioab˘a, D. Cvetkovi´c, S. M. Fallat, C. Godsil, W. Haemers, L. Hogben, R. Mikkel son, S. Narayan, O. Pryporova, I. Sciriha, W. So, D. Stevanovi´c, H. van der Holst, K. Vander Meulen, A. W. Wehe, Zero forcing sets and the minimum rank of graphs, Linear Algebra Appl., 428, (2008), 1628–1648.
 
4. F. Barioli, W. Barrett, S. Fallat, H. T. Hall, L. Hogben, H. van der Holst and B. Shader, Zero forcing parameters and minimum rank problems, Linear Algebra Appl., 433 (2010), 401–411.
 
5. F. Barioli, W. Barrett, S. Fallat, H. T. Hall, L. Hogben, B. Shader, P. van den Driessche and H. van der Holst, Parameters related to tree-width, zero forcing and maximum nullity of a graph, J. Graph Theory, 72 (2013), 146–177. 
 6. F. Barioli, S. Fallat, D. Hershkowitz, H. T. Hall, L. Hogben, H. van der Holst and B. Shader, On the minimum rank of not necessarily symmetric matrices: a preliminary study, Electron. J. Linear Algebra, 18 (2009), 126–145.
 
7. A. Berman, S. Friedland, L. Hogben, U. G. Rothblum and B. Shader, An upper bound for the minimum rank of a graph, Linear Algebra Appl., 429(7) (2008), 1629–1638.
 
8. D. Burgarth, S. Bose, C. Bruder and V. Giovannetti, Local controllability of quantum networks, phys. Rev. A, 79 (2009), Article ID: 060305.
 
9. D. Burgarth and V. Giovannetti, Full control by locally induced relaxation, phys. Rev. Lett., 99 (2007), Article ID: 100501.
 
10. S. Butler and M. Young, Throttling zero forcing propagation speed on graphs, Australas. J. Combin., 57 (2013), 65–71.
 
11. R. Davila, T. Kalinowski, and S. Stephen, A lower bound on the zero forcing number, Discrete Appl. Math., 250 (2018), 363–367.
 
12. R. Davila and F. Kenter, Bounds for the zero forcing number of graphs with large girth, Theory Appl. Graphs, 2 (2015), 1–8.
 
13. C. J. Edholm, L. Hogben, M. Huynh, J. LaGrange and D. D. Row, Vertex and edge spread of zero forcing number, maximum nullity, and minimum rank of a graph, Linear Algebra Appl., 436(12) (2012), 4352–4372.
 
14. L. Eroh, C. X. Kang and E. Yi, A comparison between the metric dimension and zero forcing number of trees and unicyclic graphs, Acta Math. Sin. (Engl. Ser.), 33(6) (2017), 731–747.
 
15. S. Fallat and L. Hogben, The minimum rank of symmetric matrices described by a graph: a survey, Linear Algebra Appl., 426 (2007), 558–582.
 
16. M. Gentner, L. D. Penso, D. Rautenbach, U. S. Souza, Extremal values and bounds for the zero forcing number, Discrete Appl. Math., 214 (2016), 196–200.
 
17. T. Kalinowski, N. Kamcev and B. Sudakov, The zero forcing number of graphs, SIAM J. Discrete Math., 33(1) (2019), 95–115.
 
18. M. Khosravi, S. Rashidi and A. Sheikhhosseni, Connected Zero Forcing Sets And Connected Propagation Time Of Graphs, Trans. Comb., 9(2) (2020), 77–88.
 
19. M. M. C. Lunar and R. A. Robles, Characterization and structure of a power set graph, International Journal of Advanced Research and Publications, 3(6) (2019), 1–4.
 
20. E. Mokhtarian Dehkordi, M. Eshaghi, S. Moradi and Z. Yarahmadi, Directed Power Graphs, International journal of nonlinear analysis and applications, 12(2) (2021), 1371–1382.
 
21. Z. Rameh and E. Vatandoost, Some Cayley graphs with propagation time 1, Journal of the Iranian Mathematical Society, 2(2) (2021), 111–122.
 
22. A. Treier, Universal graph powerset, Conference Series, Sobolev Institute of Mathematics of SB RAS, Omsk, Russia, November (2019), 1–5.
 
23. E. Vatandoost and Y. Golkhandy Pour, On the zero forcing number of some Cayley graphs, Algebraic Structures and Their Applications, 4(2) (2017), 15–25. 
Journal of Algebraic Systems
Volume 13, Issue 3
November 2025
Pages 15-27

Files

Share

How to cite

Statistics