Volume 67 | Issue 7 | Year 2021 | Article Id. IJMTT-V67I7P512 | DOI : https://doi.org/10.14445/22315373/IJMTT-V67I7P512

The distance polynomial of G is defined as the determinant |µI-D|, where I is the unit matrix of the order same as that of D. The distance spectra of a connected graph G is the collection of distance eigenvalues of G and the distance energy of G is the absolute sum of the distance eigenvalues of G. In this paper, the distance spectra and the distance energy of the graph obtained by deleting the edges of complete subgraph Kr from the complete graph Kp are obtained.

[1] D. M. Cvetkovi´c, M. Doob and H. Sachs, Spectra of Graphs.Theory and Applications, Academic Press, New York – London, (1980).

[2] K. Balasubramanian, Computer Generation of Distance Polynomials of Graphs, Journal of Computational Chemistry , 11 (1990) 829–836.

[3] I. Gutman, The energy of a graph: Old and new results, in: A. Betten, A. Kohnert, R. Laue, A. Wassermann(Eds.), Algebraic Combinatorics and Applications, Springer–Verlag, Berlin, 001(196–211).

[4] P. W. Fowler, G. Caporossi and P. Hansen, Distance matrices, Wiener indices, andrelated invariants of fullerenes, J. Phys. Chem. A. 105 (2001) 6232–6242.

[5] F. Buckley and F. Harary, Distance in Graphs, Addison Wesley, (1990).

[6] G. Indulal, I. Gutman and A. Vijayakumar, On distance energy of graphs, MATCHCommun. Math. Comput. Chem. 60 (2008) 461–472.

[7] G. Indulal and I. Gutman, On the distance spectra of some graphs, Math. Commun. 13 (2008) 123–131.

[8] R. Laskar, Eigenvalues of the adjacency matrix of cubic lattice graphs, Pacific J. Math. 29(1969) 623–629.

[9] M. Marcus and H. Mine, A survey of Matrix theory and Matrix inequalities, Allynand Bacon, Inc., Boston, Mass., 1964.

[10] B. D. McKay, On the spectral characterization of trees, Ars Combin. 3 (1977) 219–232.

[11] H. S. Ramane, I. Gutman, D. S. Revankar, Distance Equienergetic graphs, MATCH Commun. Math. Comput. Chem 60, (2008) 473-484

[12] D. Stevanovi´c and G. Indulal, The distance spectrum and energy of the compositions of regular graphs, Appl. Math. Lett. 22 (2009) 1136–1140.

[13] P. R. Hampiholi and B. S. Durgi, Characteristic polynomial of some cluster graphs, Kragujevac Journal of Mathematics, 37(2) (2013) 369 - 373.

[14] P. R. Hampiholi, H. B. Walikar and B. S. Durgi, Energy of complement of stars, J. Indones Math. Soc., Vol 19, No. 1 (2013) 15-21.

[15] P. R. Hampiholi and B. S. Durgi, On the Spectra and Energy of some cluster graphs, Int. J. Math. Sci. &Engg.Appls. (IJMSEA) ISSN 0973-9424, 7(2) (2013) 219-228

[16] B. S. Durgi, P. R. Hampiholi and S. M. Mekkalike, Distance spectra and Distance energy of some cluster graphs, Mathematica Aeterna4(8) (2014) 817 – 825

[17] H. S. Ramane, D. S. Revankar, I. Gutman, S. B. Rao, B. D. Acharya, H. B. walikar, Bounds for Distance energy of a graph, Kragujevac Journal of Mathematics, (31)(2008) 59-68.

[18] Sudhir R. Jog, Prabhakar R. Hampiholi, Anjana S. Joshi, On the energy of some graphs, Annals of Pure and Applied Mathematics, 17(1) (2018) 15 – 21.

B. S. Durgi, Umesh Poojari, D.S.Revankar, P. R. Hampiholi, "Distance Polynomial, Distance Spectra And
Distance Energy Of Some Edge Deleted Graphs," *International Journal of Mathematics Trends and Technology (IJMTT)*, vol. 67, no. 7, pp. 94-103, 2021. *Crossref*, https://doi.org/10.14445/22315373/IJMTT-V67I7P512