IJAM: Volume 37, No. 6 (2024)

DOI: 10.12732/ijam.v37i6.3

INCIDENCE ENERGY OF SUBGRAPH

COMPLEMENTS OF GRAPHS

K.V. Madhumitha, A. Harshitha,

Sabitha D’Souza, Swati Nayak

Department of Mathematics

Manipal Institute of Technology

Manipal Academy of Higher Education

Manipal, 576104, Karnataka, INDIA

Abstract. Incidence energy of the graph G, denoted by IE(G), is defined as the sum of the singular values of its incidence matrix. The notion of incidence energy of subgraph complements of a graph has been proposed in this study. The expression for the sum of singular values and eigenvalues of $[I(G \oplus S)]$ and $[I(G \oplus S)][I(G \oplus S)]^T$, respectively has been obtained. Additionally, we have noted the changes in the trace of $[I(G \oplus S)][I(G \oplus S)]^T$ upon deleting an edge of $G \oplus S$. We have characterized incidence energy of subgraph complement of $P_n$ and $K_{1,n-1}$ and also, obtained some bounds for incidence energy of subgraph

complements of a graph. The incidence energy of subgraph complement of complete graph, complete bipartite graph and star has been computed.

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How to cite this paper?
DOI: 10.12732/ijam.v37i6.3
Source: International Journal of Applied Mathematics
ISSN printed version: 1311-1728
ISSN on-line version: 1314-8060
Year: 2024
Volume: 37
Issue: 6

References

[1] D. M. Cvetkovic, M. Doob, H. Sachs, Spectra of Graphs Theory and Application, Johann Ambrosius Barth 3rd rev. and EN ed. (1980).

[2] D. S. Mitrinovic, P. M. Vasic, Analytic Inequalities, Springer (1970).

[3] F. V. Fomin, P. A. Golovach, T. J. Stromme, D. M. Thilikos, Subgraph complementation, Algorithmica, 82, No 7 (2020), 1859-1880; https://doi.org/10.1007/s00453-020-00677-8.

[4] I. Gutman, The energy of a graph, Ber. Math. Stat. Sekt. Forschungsz. Graz., 103 (1978), 1-22.

[5] I. Gutman, D. Kiani, M. Mirzakhah, On incidence energy of graphs, MATCH Commun. Math. Comput. Chem., 62, No 3 (2009), 573–580.

[6] J. Zhang, J. Li, New results on the incidence energy of graphs, MATCH Commun. Math. Comput. Chem., 68, No 3 (2012), 777–803.

[7] K.C. Das, I. Gutman, On incidence energy of graphs, Linear Algebra and Its Applications, 446, (2014), 329–344; https://doi.org/10.1016/j.laa.2013.12.026.

[8] L. Shi, H. Wang, The Laplacian incidence energy of graphs, Linear Algebra and its Applications, 439, No 12 (2013), 4056–4062; https://doi.org/10.1016/j.laa.2013.10.028.

[9] M. Jooyandeh, D. Kiani, M. Mirzakhah, Incidence energy of a graph, MATCH Commun. Math. Comput. Chem., 62, No 3 (2009), 561-572.

[10] M. Kamiґnski, V. V. Lozin, M. Milani.c, Recent developments on graphs of bounded clique-width, Discrete Applied Mathematics, 157, No 12 (2009), 2747–2761; http://dx.doi.org/10.1016/j.dam.2008.08.022.

[11] M. Matejiґc, S. B. B. Altinda.g, E. Milovanoviґc, I. Milovanoviґc, On the Randic incidence energy of graphs, Computational and Applied Mathematics, 40 (2021), 1-11; http://dx.doi.org/10.1007/s40314-021-01589-1.

[12] O. Rojo, E. Lenes, A sharp upper bound on the incidence energy of graphs in terms of connectivity, Linear Algebra and its Applications, 438, No 3 (2013), 1485–1493; https://doi.org/10.1016/j.laa.2012.09.012.

[13] R. Balakrishnan, The energy of a graph, Linear Algebra and its Applications, 387 (2004), 287-295; https://doi.org/10.1016/j.laa.2004.02.038.

[14] R. Gu, F. Huang, X. Li, Randiґc Incidence Energy of Graphs, arXiv Preprint arXiv:1405.7498. 2014 May 29 (2021).

[15] S. B. Bozkurt, D. Bozkurt, On incidence energy, MATCH Commun. Math. Comput. Chem., 72, No 2 (2014), 215–225.

[16] S. D’Souza, H.J. Gowtham, S. Nayak, P.G. Bhat, Label incidence energy of partial edge labeled graph, In: Proceedings of the Jangjeon Mathematical Society, 24, No 1 (2021), 113-123; http://dx.doi.org/10.17777/pjms2021.24.1.113.

[17] S. D’Souza, S. Nayak, P. G. Bhat, Laplacian energy of partial complement of a graph, Materials Today: Proceedings, 54, (2022), 827-831; https://doi.org/10.1016/j.matpr.2021.11.109.

[18] S. Nayak, S. D’Souza, H. J. Gowtham, P. G. Bhat, Energy of partial complements of a graph, In: Proceedings of the Jangjeon Mathematical Society, 22, No 3 (2019), 369-379; http://dx.doi.org/10.17777/pjms2019.22.3.369.

[19] V. Nikiforov, The energy of graphs and matrices, Journal of Mathematical Analysis and Applications, 326, No 2 (2007), 1472-1475; https://doi.org/10.1016/j.jmaa.2006.03.072.

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