D-Spectrum and D-Energy of Complements of Iterated Line Graphs of Regular Graphs

Document Type : Research Paper

Author

Department of Mathematics, St.Aloysius College, Edathua, Alappuzha, India

10.29252/asta.4.1.51

Abstract

The D-eigenvalues {µ1,…,µp} of a graph G are the eigenvalues of its distance matrix D and form its D-spectrum. The D-energy, ED(G) of G is given by ED (G) =∑i=1p |µi|. Two non cospectral graphs with respect to D are said to be D-equi energetic if they have the same D-energy. In this paper we show that if G is an r-regular graph on p vertices with 2r ≤ p - 1, then the complements of iterated line graphs of G are of diameter 2 and that ED(\overline{Lk(G)}), k≥2 depends only on p and r. This result leads to the construction of regular D-equi energetic pair of graphs.

Keywords


[1] M. Aouchiche and P. Hansen, Distance spectra of graphs: A survey, Lin. Algebra Appl. 458(2014), 301-386.
[2] B. Bollobas, The diameter of random graphs, Trans. Amer. Math. Soc. 267 (1981), 41-52.
[3] F. Buckley and F. Harary, Distance in Graphs, Addison Wesley, Redwood. 1990.
[4] F. Buckley, The size of iterated line graphs, Graph Theory Notes New York. 25 (1993), 33-36.
[5] D.M. Cvetkovic, M. Doob and H. Sachs, Spectra of Graphs-Theory and Applications, Academic Press, (1980).
[6] M. Edelberg, M.R. Garey and R.L. Graham, On the distance matrix of a tree, Discrete Math. 14 (1976), 23-39.
[7] F.R. Gantmacher, Applications of the Theory of Matrices (Interscience, New York, 1959), Google Schola. (1969), p.64.
[8] R.L. Graham and L. Lovasz, Distance matrix polynomials of trees. Adv Math. 29 (1978), 60-88.
[9] I. Gutman, On graphs whose energy exceeds the number of vertices, Lin. Algebra Appl. 429 (11-12) (2008), 2670-2677.
[10] I. Gutman, S. Zare Firoozabadi, J.A. delaPena and J. Rada, On the energy of regular graphs, Match Commun. Math. Comput. Chem. 57 (2007), 435-442.
[11] W.H. Haemers, Strongly regular graphs with maximal energy, Lin. Algebra Appl. 429 (11-12) (2008), 2719-2723.
[12] G. Indulal, I. Gutman and A. Vijayakumar, On distance energy of graphs, Match Commun. Math. Comput. Chem. 60 (2008), 461- 472.
[13] G. Indulal and I. Gutman, On the distance spectra of somegraphs, Math. Commun. 13 (2008), 123-131.
[14] G. Indulal and A. Vijayakumar, A note on energy of some graphs, Match Commun. Math. Comput. Chem. 59 (2008), 269-274.
[15] X. Li and J. Zhang, On bicyclic graphs with maximal energy, Lin. Algebra Appl. 427 (2007), 87-98.
[16] J.W. Moon and L. Moser, Almost all (0; 1) matrices are primitive, Studia Sci Math Hungar.1(1966), 153-156.
[17] V. Nikiforov, The energy of graphs and matrices, J. Math. Anal. Appl. 326 (2007), 1472-1475.
[18] I. Shparlinski, On the energy of some circulant graphs, Lin. Algebra Appl. 414 (2006), 378-382.