DISTANCE SPECTRA OF SOME GRAPH OPERATIONS AND SOME NEW DISTANCE EQUIENERGETIC GRAPHS OF DIAMETER 3

Year 2019, Volume: 9 Issue: 3, 581 - 588, 01.09.2019

C. Adiga B. R. Rakshith - Sumithra

Abstract

Two graphs of same order are said to be distance equienergetic if their distance energies are same. In this paper, we rst give a partial insight on the distance spectrum of Mycielskian graphs and then we focus on constructing distance equienergetic graphs by introducing three new graph operations. As an application of our results, we construct some new class of distance equinergetic graphs of diameter 3 on 18+2n vertices for all n 1.

Keywords

Distance matrix, distance equienergetic graphs, Mycielskian graphs.

References

• Aalipour, G., Abiad, A., Berikkyzy, Z., Cummings, J., De Silva, J., Gao, W., Heysse, K., Hogben, L., Kenter, F. H. J., Lin, J. C.-H., Tait, M., (2016), On the distance spectra of graphs, Linear Algebra Appl., 497, pp. 66–87.
• Adiga, C., Rakshith, B. R., (2016), On spectra of variants of the corona of two graphs and some new equienergetic graphs, Discussions Math. Graph Thy., 36, pp. 127–140.
• Adiga, C., Balakrishnan, R., So, W., (2010), The skew energy of a digraph, Linear Algebra Appl., , pp. 1825–1835.
• Adiga, C., Rakshith, B. R., (2016), More skew–equienergetic digraphs, Commun. Comb. Opt., 1, pp. –66.
• Aouchiche, M., Hansen, P., (2014), Distance spectra of graphs: a survey, Linear Algebra Appl., 458, pp. 301–386.
• Balakrishnan, R., (2004), The energy of a graph, Linear Algebra Appl., 387, pp. 287–295.
• Balakrishnan, R., Kavaskar, T., So, W., (2012), The energy of the Mycielskian of a regular graph
• Australasian J. Comb., 52, pp. 163–171. Behtoei, A., Anbarloei, M., (2016), Degree Distance Index of the Mycielskian and its Complement
• Iranian J. Math. Chem., 7, pp. 1–9. Brankov, V., Stevanovi´c, D., Gutman, I., (2004), Equienergetic chemical trees, J. Serb. Chem. Soc., , pp. 549–553.
• Consonni, V., Todeschini, R., (2008), New spectral indices for molecule description, MATCH Commun.
• Math. Comput. Chem., 60, pp. 3–14. Cvetkovi´c, D., Doob, M., Sachs, H., (1995), Spectra of Graphs - Theory and Application, Johann
• Ambrosius Barth Verlag. Ferrar, W. L., (1953), A Text-Book of Determinants, Matrices and Algebraic Forms, Oxford University Press.
• Gong, S., Li, X., Xu, G., Gutman, I., Furtula, B., (2015), Borderenergetic graphs, MATCH Commun.
• Math. Comput. Chem., 74, pp. 321–332. Graham, R. L., Pollack, H. O., (1971), On the addressing problem for loop switching, Bell Syst. Tech. J., 50, pp. 2495-2519.
• Gutman, I., (1978), The energy of a graph, Ber. Math. Statist. Sekt. Forschungsz. Graz, 103, pp. –22.
• Ili´c, A., (2010), Distance spectra and distance energy of integral circulant graphs, Linear Algebra Appl., 433, pp. 1005–1014.
• Indulal, G., Gutman, I., Vijayakumar, A., (2008), On distance energy of graphs, MATCH Commun.
• Math. Comput. Chem., 60, pp. 461–472. Indulal, G., Stevanovi´c, D., (2015), The distance spectrum of the corona and cluster of two graphs
• AKCE Int. J. Graphs Comb., 12, pp. 186-192. Indulal, G., Balakrishnan, R., (2016), Distance spectrum of Indu–Bala product of graphs, AKCE Int. J. Graphs Comb., 3, pp. 230–234.
• Li, X., Shi, Y., Gutman, I., (2012), Graph Energy, Springer, New York.
• Mycielski, J., (1955), Sur le coloriage des graphs, Colloq. Math., 3, pp. 161–162.
• Ramane, H. S., Gutman, I., Revankar, D. S., (2008), Distance equienergetic graphs, MATCH Com- mun. Math. Comput. Chem., 60, pp. 473–484.
• Ramane, H. S., Revankar, D. S., Gutman, I., Walikar, H. B., (2009), Distance spectra and distance energies of iterated line graphs of regular graphs, Publ. Inst. Math., 85, pp. 39–46.
• Ramane, H. S., Walikar, H. B., Rao, S. B., Acharya, B. D., Hampiholi, P. R., Jog, S. R., Gutman, I., (2004), Equienergetic graphs, Kragujevac J. Math., 26, pp. 5–13.
• Stevanovi´c, D., Indulal, G., (2009), The distance spectrum and energy of the compositions of regular graphs, Appl. Math. Lett., 22, pp. 1136–1140.
• Xu, L., Hou, Y., (2007), Equienergetic bipartite graphs, MATCH Commun. Math. Comput. Chem., , pp. 363–370.