Null Spaces Dimension of the Eigenvalue -1 in a Graph

Authors

  • Gohdar H. Mohiaddin Department of Mathematic, Faculty of Science, University of Zakho, Kurdistan Region, Iraq
  • Khidir R. Sharaf Department of Mathematics, Faculty of Science, University of Zakho, Kurdistan Region, Iraq

DOI:

https://doi.org/10.25271/sjuoz.2019.7.4.609

Keywords:

graph theory, high zero sum weighting, adjacency matrix, nullity, corona product

Abstract

In geographic, the eigenvalues and eigenvectors of transportation network provides many informations about its connectedness. It is proven that the more highly connected in a transportation network G has largest eigenvalue and hence more multiple occurrences of the eigenvalue -1. For a graph G with adjacency matrix A, the multiplicity of the eigenvalue -1 equals the dimension of the null space of the matrix A + I. In this paper, we constructed a high closed zero sum weighting of G and by which its proved that, the dimension of the null space of the eigenvalue -1 is the same as the number of independent variables used in a non-trivial high closed zero sum weighting of the graph. Multiplicity of -1 as an eigenvalue of known graphs and of corona product of certain classes of graphs are determined and two classes of -1- nut graphs are constructed.

Author Biographies

Gohdar H. Mohiaddin, Department of Mathematic, Faculty of Science, University of Zakho, Kurdistan Region, Iraq

Dept. of Mathematic, Faculty of Science, University of Zakho, Kurdistan Region, Iraq - (gohdar.mohiaddin@uoz.edu.krd

Khidir R. Sharaf, Department of Mathematics, Faculty of Science, University of Zakho, Kurdistan Region, Iraq

Dept. of Mathematics, Faculty of Science, University of Zakho, Kurdistan Region, Iraq - (khidir.sharaf)@uoz.edu.krd

References

Ali , D.A., Gauci, J.B., Sciriha, I., & Sharaf, K.R. (2016a). Coalescing Fiedler and Core Vertices. Czechoslovak Mathematical 971-985.
Ali , D.A., Gauci, J.B., Sciriha, I., & Sharaf, K.R. (2016b). Nullity of a Graph with a Cut-edge. Communication in Mathematical and in Computer Chemistry, 76, 771-791.
Ali, D.A., Gauci, J.B., Sciriha, I., & Sharaf, K.R. (2019). The Conductivity of Superimposed Key-graphs with a Common One Dimensional Adjacency Nullspace. ARS Mathematica Contemporanea, 16, 1-15.
Brouwer , A. E., & Haemers, W.H. (2011). Spectra of Graphs. Springer.
Cheng, B., & Liu, B. (2007). On the Nullity of graphs. Journal of Linear Algebra,16.
Cvetkovic, D. M., Doob, M., & Sachs, H. (New York 1979). Spectra of Graphs-Theory and Applications. Acamemic Press.
Mohan, S., Geetha, J., & Somasundaram,KB. (2017). Total Coloring of the Corona Product of two Graphs. Australasian Journal of Combinatorics, 68.
Mohiaddin, G.H., & Sharaf, K.R. (2018). Construction and Nullity of Some Classes of Smith Graphs. IEEE Xplore.
Sciriha, I. (2007). A characterization of singular graphs. Electronic Journal of Linear Algebra, 16, 451-462.
Sharaf, K.R., & Rashed, P.A. (2002). On the Degree of the Singular of a Graph. J. of Duhok University, (5), 133-138.

Downloads

Published

2019-12-30

How to Cite

Mohiaddin, G. H., & Sharaf, K. R. (2019). Null Spaces Dimension of the Eigenvalue -1 in a Graph. Science Journal of University of Zakho, 7(4), 167–171. https://doi.org/10.25271/sjuoz.2019.7.4.609

Issue

Vol. 7 No. 4 (2019): 30, December

Section

Science Journal of University of Zakho

License

Authors who publish with this journal agree to the following terms:

  • Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License [CC BY-NC-SA 4.0] that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal.
  • Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work, with an acknowledgment of its initial publication in this journal.
  • Authors are permitted and encouraged to post their work online.