# Þ-energy of generalized Petersen graphs

Document Type : Original paper

Authors

1 Research scholar, Department of mathematics, CHRIST (Deemed to be University), Bangalore, India

2 Department of mathematics, Faculty, CHRIST (Deemed to be University), Bangalore.

Abstract

For a given graph $G$, its $\mathscr{P}$-energy is the sum of the absolute values of the eigenvalues of the  $\mathscr{P}$-matrix of $G$. In this article, we explore the $\mathscr{P}$-energy of generalized Petersen graphs $G(p,k)$ for various vertex partitions such as independent, domatic, total domatic and $k$-ply domatic partitions and partition containing a perfect matching in $G(p,k)$. Further, we present a python program to obtain the $\mathscr{P}$-energy of $G(p,k)$ for the vertex partitions under consideration and examine the relation between them.

Keywords

Main Subjects

#### References

[1] R. Balakrishnan and K. Ranganathan, A Textbook of Graph Theory, Springer Science & Business Media, 2012.
[2] E.J. Cockayne, R.M. Dawes, and S.T. Hedetniemi, Total domination in graphs, Networks 10 (1980), no. 3, 211–219.
[3] E.J. Cockayne and S.T. Hedetniemi, Towards a theory of domination in graphs, Networks 7 (1977), no. 3, 247–261.
[4] H.S.M. Coxeter, Self-dual configurations and regular graphs, Bull. Am. Math. Soc. 56 (1950), no. 5, 413–455.
[5] D.M. Cvetković, M. Doob, and H. Sachs, Spectra of Graphs- Theory and Application, Academic Press, New York, 1980.
[6] M. Farber, G. Hahn, P. Hell, and D. Miller, Concerning the achromatic number of graphs, J. Combin. Theory, Ser. B 40 (1986), no. 1, 21–39.
[7] I. Gutman, The energy of a graph, Ber. Math. Stat. Sekt. Forschungsz. Graz. 103 (1978), 1–22.
[8] I. Gutman, X. Li, and J. Zhang, Graph Energy, Springer, New York, 2012.
[9] S. Hedetniemi, P. Slater, and T.W. Haynes, Fundamentals of Domination in Graphs, Marcel De. Inc., New York, 1998.
[10] P.B. Joshi and M. Joseph, P-energy of graphs, Acta Univ. Sapientiae, Info. 12 (2020), no. 1, 137–157.
[11] , New results on P-energy of join of graphs, Malaya J. Matematik 8 (2020), no. 4, 2092–2096.
[12] , On P-energy of join of graphs, Malaya J. Matematik 8 (2020), no. 4, 2082–2087.
[13] T. Moscibroda and R. Wattenhofer, Maximizing the lifetime of dominating sets, 19th IEEE International Parallel and Distributed Processing Symposium, IEEE, 2005, pp. 8–pp.
[14] S.V. Pemmaraju and I.A. Pirwani, Energy conservation via domatic partitions, Proceedings of the 7th ACM international symposium on Mobile ad hoc networking and computing, 2006, pp. 143–154.
[15] T. Pisanski and M. Krnc, Generalized petersen graphs and kronecker covers, Discrete Math. Theor. Comput. Sci. 21 (2019), 1–16.
[16] E. Sampathkumar, S.V. Roopa, K.A. Vidya, and M.A. Sriraj, Partition energy of a graph, Proc. Jangjeon Math. Soc. 18 (2015), no. 4, 473–493.
[17] M.E. Watkins, A theorem on tait colorings with an application to the generalized petersen graphs, J. Comb. Theory 6 (1969), no. 2, 152–164.
[18] D.B. West, Introduction to Graph Theory, Pearson, New Jersey, 2001.
[19] B. Zelinka, On k-ply domatic numbers of graphs, Math. Slovaca 34 (1984), no. 3, 313–318.
[20] C. Zhihao, A note on symmetric block circulant matrix, J. Math. Res. Exposition, vol. 10, 1990, pp. 469–473.
[21] A. Zitnik, B. Horvat, and T. Pisanski, All generalized petersen graphs are unitdistance graphs, J. Korean Math. Soc. 49 (2012), no. 3, 475–491.