Nullity of b-Bridge Coalescence Graphs

Khidir R. Sharaf and Nechirvan B. Ibrahim
Department of Mathematics, Faculty of Science, University of Zakho, Zakho, Kurdistan Region, Iraq.


Abstract

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.

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).