Null Spaces Dimension of the Eigenvalue -1 in a Graph

  • 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
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

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Published
2019-10-30
How to Cite
Mohiaddin, G., & Sharaf, K. (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
Section
Science Journal of University of Zakho