Khidir R. Sharaf and Nechirvan B. Ibrahim

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

The nullity η(G) (degree of singularity) of a graph G is the algebraic multiplicity of the

number zero in the spectrum of G. If G is a graph containing a vertex of degree one and

H be the subgraph obtained from G, by deleting this vertex together with the vertex

adjacent to it then, η(G) = η(H). In this paper, we proved that nullity of a graph is the

maximum number of independent variables in a high zero-sum weighting for it. The

above procedures are applied to evaluate the nullity of b-bridge coalescence graphs.

They are also applied to determine the nullity of edge introducing between t-tuple

coalescence graphs and nullity of paths introducing between (n, m)-tuples of coalescence

graphs.

[1] Ali A. M. and Dahash N.M.; Hosoya polynomials of coalescence and bridges coalescence graphs, The 4th Int. Conference on Information Technology, Jordan. (2013).

[2] Ali D .E.; On the Nullity of Coalescence of Some Graphs, M.Sc. Thesis, University of Duhok, Iraq. (2010).

[3] Bondy J. A. and Murty U.S.R.; Graph Theory, Graduate Texts in Mathematics, Vol. 244, Springer-verlag, New York. (2008).

[4] Borovic¢anin B. and Gutman I; Nullity of graphs, Matematicki Institute Sanu, Belgrade, pp.107-122. (2000).

[5] Brown M., Kennedy J.W. and Servatius B.; Graph singularity, Graph Theory Notes of New York, XXV, pp.23-32. (1993).

[6] Cheng B. and Liu, B.; On the nullity of graphs, J. of Linear Algebra, Vol. 16, pp.60-67. (2007).

[7] Cvetkovic¢ D. M., Doob M. and Sachs H.; Spectra of Graphs-Theory and Application, Academic Press, New York. (1979).

[8] Gong S. C. and Xu, G. H.; On the nullity of a graph with cut-points ,Linear Algebra and its Applications, Vol.436,pp.135–142, (2012).

[9] Ibrahim N. B.; On the Nullity of Some Sequential Element Identified, Element Introduced Graphs, M.Sc. Thesis, University of Zakho, Iraq. (2013). [10] Mohiaddin G. H.; On the Nullity of Some Graphs, M.Sc. Thesis, University of Zakho, Iraq. (2011).

**Abstract**

number zero in the spectrum of G. If G is a graph containing a vertex of degree one and

H be the subgraph obtained from G, by deleting this vertex together with the vertex

adjacent to it then, η(G) = η(H). In this paper, we proved that nullity of a graph is the

maximum number of independent variables in a high zero-sum weighting for it. The

above procedures are applied to evaluate the nullity of b-bridge coalescence graphs.

They are also applied to determine the nullity of edge introducing between t-tuple

coalescence graphs and nullity of paths introducing between (n, m)-tuples of coalescence

graphs.

**Key Words:**

Spectra of a Graph, Nullity of a Graph, b- Bridge Graphs.

**References**

[1] Ali A. M. and Dahash N.M.; Hosoya polynomials of coalescence and bridges coalescence graphs, The 4th Int. Conference on Information Technology, Jordan. (2013).

[2] Ali D .E.; On the Nullity of Coalescence of Some Graphs, M.Sc. Thesis, University of Duhok, Iraq. (2010).

[3] Bondy J. A. and Murty U.S.R.; Graph Theory, Graduate Texts in Mathematics, Vol. 244, Springer-verlag, New York. (2008).

[4] Borovic¢anin B. and Gutman I; Nullity of graphs, Matematicki Institute Sanu, Belgrade, pp.107-122. (2000).

[5] Brown M., Kennedy J.W. and Servatius B.; Graph singularity, Graph Theory Notes of New York, XXV, pp.23-32. (1993).

[6] Cheng B. and Liu, B.; On the nullity of graphs, J. of Linear Algebra, Vol. 16, pp.60-67. (2007).

[7] Cvetkovic¢ D. M., Doob M. and Sachs H.; Spectra of Graphs-Theory and Application, Academic Press, New York. (1979).

[8] Gong S. C. and Xu, G. H.; On the nullity of a graph with cut-points ,Linear Algebra and its Applications, Vol.436,pp.135–142, (2012).

[9] Ibrahim N. B.; On the Nullity of Some Sequential Element Identified, Element Introduced Graphs, M.Sc. Thesis, University of Zakho, Iraq. (2013). [10] Mohiaddin G. H.; On the Nullity of Some Graphs, M.Sc. Thesis, University of Zakho, Iraq. (2011).