Harary energy of complement of line graphs of regular graphs

Year 2020, Volume: 69 Issue: 2, 1215 - 1220, 31.12.2020

Harishchandra Ramane , K. Ashoka

https://doi.org/10.31801/cfsuasmas.630087

Abstract

The Harary matrix of a graph $G$ is defined as $H(G) = [h_{ij}]$, where $h_{ij} =\frac{1}{d(v_i, v_j)}$, if $i \neq j$ and $h_{ij} = 0$, otherwise, where $d(v_i, v_j)$ is the distance between the vertices $v_i$ and $v_j$ in $G$. The $H$-energy of $G$ is defined as the sum of the absolute values of the eigenvalues of Harary matrix. Two graphs are said to be $H$-equienergetic if they have same $H$-energy. In this paper we obtain the $H$-energy of the complement of line graphs of certian regular graphs interms of the order and regularity of a graph and thus constructs pairs of $H$-equienergetic graphs of same order and having different $H$-eigenvalues.

Keywords

Harary eigenvalues, energy, equienergetic graphs

Supporting Institution

University Grants Commission (UGC), New Delhi

Project Number

F.510/3/ DRS-III /2016 (SAP-I)

Thanks

The author HSR is thankful to the University Grants Commission (UGC), New Delhi, for support through UGC-SAP DRS-III, 2016-2021: F.510/3/ DRS-III /2016 (SAP-I).

References

• Buckley, F., Iterated line graphs, Congr. Numer., 33 (1981), 390--394.
• Buckley, F., The size of iterated line graphs, Graph Theory Notes New York, 25 (1993), 33--36.
• Cui, Z., Liu, B., On Harary matrix, Harary index and Harary energy, MATCH Commun. Math. Comput. Chem., 68 (2012), 815-823.
• Cvetković, D., Rowlinson, P., Simić, S., An Introduction to the Theory of Graph Spectra, Cambridge Univ. Press, Cambridge, 2010.
• Güngör, A. D., Çevik, A. S., On the Harary energy and Harary Estrada index of a graph, MATCH Commun. Math. Comput. Chem., 64 (2010), 280-296.
• Gutman, I., The energy of a graph, Ber. Math. Stat. Sekt. Forschungsz. Graz, 103 (1978), 1-22.
• Harary, F., Graph Theory, Addison--Wesley, Reading, 1969.
• Indulal, G., D-spectrum and D-energy of complements of iterated line graphs of regular graphs, J. Alg. Stru. Appl., 4 (2017), 51-56.
• Ivanciuc, O., Balaban, T. S., Balaban A. T., Design of topological indices. Part 4. Reciprocal distance matrix, related local vertex invariants and topological indices, J. Math. Chem., 12 (1993), 309-318.
• Li, X., Shi Y., Gutman, I., Graph Energy, Springer, New York, 2012.
• Plavšić, D., Nikolić, S., Trinajstić N., On the Harary index for the characterization of chemical graphs, J. Math. Chem., 12 (1993), 235-250.
• Ramane, H. S., Manjalapur V. V., Harary equienergetic graphs, Int. J. Math. Arch., 6 (2015), 81-86.
• Ramane, H. S., Jummannaver, R. B., Harary spectra and Harary energy of line graphs of regular graphs, Gulf J. Math., 4 (2016), 39-46.
• Sachs, H., Über selbstkomplementare Graphen, Publ. Math. Debrecen, 9 (1962), 270-288.
• Sachs, H., Über Teiler, Faktoren und charakteristische Polynome von Graphen, Teil II, Wiss. Z. TH Ilmenau, 13 (1967), 405-412.
• Xu, K., Das, K. C., Trinajstić, N., The Harary Index of a Graph, Springer, Heidelberg, 2015.