Payman A. Rashed

Dep. of Mathematics, College of Basic Education,University of Salahaddin, Erbil, Iraq

DOI: https://doi.org/10.17656/jzs.10622

In this paper the idea of high zero-sum weighting of new technique is used to find the nullity of some special graphs Gp, Gk, Gc which are constructed by identifying the path graph pn of different order with star graph, complete graph, and cycle graph ,pn S1,m, pn Kp and pn cm, respectively. And it is proved that the nullity of these new graphs is increases or decreases according to the changing of identifying vertex in the path, or the other graphs, and with the order of path which is even or odd.

[1] B.Andrasfai, "Graph Theory, Flows and matrices", Adam Hilgar, Budapest, Hungary,( 1991).

[2] S.Axler, K.A.Ribet, "Graph Theory", Graduate texts in Mathematics, ( 1993).

[3] L.W.Beneke and R.J.Wilson, " Selected Topics in Graph Theory", Academic press, Londan,( 1978).

[4] L.W.Beneke , R.J.Wilson and P.J. Cameron,"Topics in Algebraic Graph Theory", Cambridge University press, (2004).

[5] Broun, J.W. Kenedy and B. Servatios, "Graph singularity", Graph Theory notes of New York, Vol.25, pp. 23-32,(1993).

[6] D.M. Cvetcovic, M. Doob, I. Gutman and A. Torgasev, "Resent Result in the Theory of Graph spectra", North Holland, Amsterdam, (1988).

[7] D.M. Cvetcovic , M. Doob and H.sachs, "Spectra of Graphs", Academic press, New York, (1976).

[8] D.M. Cvetcovic , M. Doob and H.sachs, "Spectra of Graphs", Theory and Applications,3 revised and enlarged edition, Johan Ambrosius Bart. Verlange Heidelberg-Leipziy, (1995).

[9] R. Faroq, M. Ali, Q. Naureen, and Sh. Pirzada, "On the nullity of a family of tripartite graphs", Acta Univ. Sapientiae, Informatica Vol. 8, Issu.1, pp. 96-107, (2016).

[10] I.Gutman and I.scriha,"On the Nullity of line Graphs of Trees", Discrete Math, Vol.232, pp.35-45, (2001).

[11] M. G. Karpovsky, "On a New Class of Codes for Identifying Vertices in Graphs", Ieee Transactions on information theory, Vol. 44, No. 2, (1998).

[12] I. Sciriha, " A characterization of singular graph", Electronic J. of Linear algebra, Vol. 16, pp. 451-462, (2007).

[13] K.R .Sheraf and D.H. Muhammed, "Degree of singularity of complement of some Graphs and Graphs with loops", J.of duhuk university, Vol. 8, No. 2, pp.101-110, (2005).

[14] K.R. Sharaf, "An upper bound for the degree of singularity of Graphs", J. of Duhok University, Vol. 7, No. 2, pp. 25-28, (2004) .

[15] Kh.R. Sharaf and P.A. Rashed, "On the degree of the singularity of a Graph", J.of Duhuk University, Vol.5, No. 2 pp.133-138, (2002).

[16] Kh.R. Sharaf and K.B. Rasul, "On the nullity of expanded graphs", Gen. Math. Nots, Vol. 21, No. 1 pp.97-117, (2014).

Dep. of Mathematics, College of Basic Education,University of Salahaddin, Erbil, Iraq

DOI: https://doi.org/10.17656/jzs.10622

**Abstract**In this paper the idea of high zero-sum weighting of new technique is used to find the nullity of some special graphs Gp, Gk, Gc which are constructed by identifying the path graph pn of different order with star graph, complete graph, and cycle graph ,pn S1,m, pn Kp and pn cm, respectively. And it is proved that the nullity of these new graphs is increases or decreases according to the changing of identifying vertex in the path, or the other graphs, and with the order of path which is even or odd.

**Key Words:**Path graph, identifying graphs, The nullity of a graph.**References**

[1] B.Andrasfai, "Graph Theory, Flows and matrices", Adam Hilgar, Budapest, Hungary,( 1991).

[2] S.Axler, K.A.Ribet, "Graph Theory", Graduate texts in Mathematics, ( 1993).

[3] L.W.Beneke and R.J.Wilson, " Selected Topics in Graph Theory", Academic press, Londan,( 1978).

[4] L.W.Beneke , R.J.Wilson and P.J. Cameron,"Topics in Algebraic Graph Theory", Cambridge University press, (2004).

[5] Broun, J.W. Kenedy and B. Servatios, "Graph singularity", Graph Theory notes of New York, Vol.25, pp. 23-32,(1993).

[6] D.M. Cvetcovic, M. Doob, I. Gutman and A. Torgasev, "Resent Result in the Theory of Graph spectra", North Holland, Amsterdam, (1988).

[7] D.M. Cvetcovic , M. Doob and H.sachs, "Spectra of Graphs", Academic press, New York, (1976).

[8] D.M. Cvetcovic , M. Doob and H.sachs, "Spectra of Graphs", Theory and Applications,3 revised and enlarged edition, Johan Ambrosius Bart. Verlange Heidelberg-Leipziy, (1995).

[9] R. Faroq, M. Ali, Q. Naureen, and Sh. Pirzada, "On the nullity of a family of tripartite graphs", Acta Univ. Sapientiae, Informatica Vol. 8, Issu.1, pp. 96-107, (2016).

[10] I.Gutman and I.scriha,"On the Nullity of line Graphs of Trees", Discrete Math, Vol.232, pp.35-45, (2001).

[11] M. G. Karpovsky, "On a New Class of Codes for Identifying Vertices in Graphs", Ieee Transactions on information theory, Vol. 44, No. 2, (1998).

[12] I. Sciriha, " A characterization of singular graph", Electronic J. of Linear algebra, Vol. 16, pp. 451-462, (2007).

[13] K.R .Sheraf and D.H. Muhammed, "Degree of singularity of complement of some Graphs and Graphs with loops", J.of duhuk university, Vol. 8, No. 2, pp.101-110, (2005).

[14] K.R. Sharaf, "An upper bound for the degree of singularity of Graphs", J. of Duhok University, Vol. 7, No. 2, pp. 25-28, (2004) .

[15] Kh.R. Sharaf and P.A. Rashed, "On the degree of the singularity of a Graph", J.of Duhuk University, Vol.5, No. 2 pp.133-138, (2002).

[16] Kh.R. Sharaf and K.B. Rasul, "On the nullity of expanded graphs", Gen. Math. Nots, Vol. 21, No. 1 pp.97-117, (2014).