Bounds on some energy-like invariants of corona and edge corona of graphs

Chinglensana Phanjoubam and Sainkupar Mn Mawiong
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 28, 2022, Number 3, Pages 383–398
DOI: 10.7546/nntdm.2022.28.3.383-398
Full paper (PDF, 199 Kb)

Details

Authors and affiliations

Chinglensana Phanjoubam
Department of Mathematics, North-Eastern Hill University
Shillong-793022, India

Sainkupar Mn Mawiong
Department of Basic Sciences and Social Sciences, North-Eastern Hill University
Shillong-793022, India

Abstract

The Laplacian-energy-like invariant of a finite simple graph is the sum of square roots of all its Laplacian eigenvalues and the incidence energy is the sum of square roots of all its signless Laplacian eigenvalues. In this paper, we give the bounds on the Laplacian-energy-like invariant and incidence energy of the corona and edge corona of two graphs. We also observe that the bounds on the Laplacian-energy-like invariant and incidence energy of the corona and edge corona are sharp when the graph is the corona or edge corona of two complete graphs.

Keywords

• Laplacian-energy-like invariant
• Incidence energy
• Corona
• Edge corona
• Ozeki’s inequality

2020 Mathematics Subject Classification

• 05C07
• 05C50
• 05C76

References

1. Barik, S., Pati, S., & Sarma, B. K. (2007). The spectrum of the corona of two graphs. SIAM Journal on Discrete Mathematics, 21(1), 47–56.
2. Chen, H., & Liao, L. (2017). The normalized Laplacian spectra of the corona and edge corona of two graphs. Linear and Multilinear Algebra, 65(3), 582–592.
3. Chen, X., Hou, Y., & Li, J. (2016). On two energy-like invariants of line graphs and related graph operations. Journal of Inequalities and Applications, 51, 1–15.
4. Consonni, V., & Todeschini, R. (2008). New spectral index for molecule description. MATCH Communications in Mathematical and in Computer Chemistry, 60, 3–14.
5. Cui, S-Y., & Tian, G-X. (2019). Some improved bounds on two energy-like invariants of some derived graphs. Open Mathematics, 17, 883–893.
6. Cvetkovíc, D., Rowlinson, P., & Simíc, S. (2010). An Introduction to the Theory of Graph Spectra. London Mathematical Society Student Texts, vol. 75, Cambridge University Press, Cambridge.
7. Das, K. C. (2003). Sharp bounds for the sum of the squares of the degrees of a graph. Kragujevac Journal of Mathematics, 25, 31–49.
8. Das, K. C. (2007). A sharp upper bound for the number of spanning trees of a graph. Graphs and Combinatorics, 23, 625–632.
9. De Caen, D. (1998). An upper bound on the sum of squares of degrees in a graph. Discrete Mathematics, 185, 245–248.
10. Frucht, R., & Harary, F. (1970). On the corona of two graphs. Aequationes Mathematicae, 4, 322–325.
11. Gutman, I. (1978). The energy of a graph. Berichte der Mathematisch-statistischen Sektion im Forschungszentrum Graz, 103, 1–22.
12. Gutman, I., Kiani, D., & Mirzakhah, M. (2009). On incidence energy of graphs. MATCH Communications in Mathematical and in Computer Chemistry, 62, 573–580.
13. Gutman, I., & Polansky, O. E. (1986). Mathematical Concepts in Organic Chemistry. Springer-Verlag, Berlin.
14. Gutman, I., & Zhou, B. (2006). Laplacian energy of a graph. Linear Algebra and Its Applications, 414, 29–37.
15. Hou, Y., & Shiu, W. C. (2010). The spectrum of the edge corona of two graphs. Electronic Journal of Linear Algebra, 20, 586–594.
16. Ilíc, A., & Stevanovíc, D. (2009). On comparing Zagreb indices. MATCH Communications in Mathematical and in Computer Chemistry, 62, 681–687.
17. Izumino, S., Mori, H., & Seo, Y. (1998). On Ozeki’s inequality. Journal of Inequalities and Applications, 2, 235–253.
18. Jooyandeh, M. R., Kiani, D., & Mirzakhah, M. (2009). Incidence energy of a graph. MATCH Communications in Mathematical and in Computer Chemistry, 62, 561–572.
19. Li, X., Shi, Y., & Gutman, I. (2010). Graph Energy. Springer, New York.
20. Liu, J., & Liu, B. (2008). A Laplacian-energy-like invariant of a graph. MATCH
    Communications in Mathematical and in Computer Chemistry, 59, 397–419.
21. Nikiforov, V. (2007). The energy of graphs and matrices. Journal of Mathematical Analysis and Applications, 326(2), 1472–1475.
22. Ozeki, N. (1968). On the estimation of the inequalities by the maximum, or minimum values. Journal of College Arts and Science, Chiba University, 5, 199–203.
23. Pirzada, S., Ganie, H. A., & Gutman, I. (2015). On Laplacian-energy-like invariant and Kirchhoff index. MATCH Communications in Mathematical and in Computer Chemistry, 73, 41–59.
24. Stevanovíc, D., Ilíc, A., Onisor, C., & Diudea, M. (2009). LEL-a newly designed molecular descriptor. Acta Chimica Slovenica, 56, 410–417.
25. Wang, S., & Zhou, B. (2013). The signless Laplacian spectra of the corona and edge corona of two graphs. Linear and Multilinear Algebra, 61(2), 197–204.
26. Wang, W., & Luo, Y. (2012). On Laplacian-energy-like invariant of a graph. Linear Algebra and Its Applications, 437, 713–721.

Manuscript history

• Received: 11 May 2021
• Revised: 29 May 2022
• Accepted: 5 July 2022
• Online First: 8 July 2022