The Laplacian Minimum Boundary Pendant Dominating Energy of a Graph
Abstract
Let $G$ be a finite, simple, and undirected graph with vertex set $V(G)$. A subset $S \subseteq V(G)$ is called a boundary pendant dominating set if the induced subgraph $<S>$ is boundary dominated by atleast one pendant vertex. The boundary pendant dominating number, denoted $\gamma^{B}_{pe}(G)$, is the minimum cardinality among all boundary pendant dominating set of $G$. In this research article, we compute the Laplacian minimum boundary pendant dominating energy $LE^{B}_{pe}(G)$ for several standard graphs, and establish corresponding upper and lower bounds for $LE^{B}_{pe}(G)$.
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References
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2. T. AleksiˆAˇtc (2008), Upper Bounds for Laplacian Energy of Graphs. MATCH Communications in Mathematics and Computer Chemistry, vol. 60, pp. 435-439.
3. R. B. Bapat and S. Pati (2004), Energy of a graph is never an odd integer, Bulletin of the Kerala Mathematical Association, vol. 1, pp. 129–132.
4. R. B. Bapat (2011), Graphs and Matrices, Hindustan Book Agency, page no. 32.
5. N. N. M. de Abreu, C. T. M. Vinagre, A. S. Bonif´acio and I. Gutman (2008), The Laplacian energy of some Laplacian integral graphs, MATCH Communications in Mathematics and in Computer Chemistry, vol. 60, pp. 447-460.
6. G. H. Fath Tabar, A. R. Ashrafi and I. Gutman (2008), Note on Laplacian energy of graphs, Bulletin de l’Acad´emie Serbe des Sciences et des Arts, Classe des Sciences Math´ematiques et Naturelles, vol. 137(33), pp. 1-10.
7. I. Gutman (1978), The Energy of a Graph, Berichte der Mathematisch-Statistischen Sektion im Forschungszentrum Graz, vol. 103, pp. 1–22.
8. I. Gutman (2001) The energy of a graph: Old and new results, In A. Betten, A. Kohnert, R. Laue, and A. Wassermann (Eds.), Algebraic Combinatorics and Applications, Springer, Berlin, pp. 196-211.
9. I. Gutman and B. Zhou (2006), Laplacian energy of a graph, Linear Algebra and its Applications, vol. 414(1), pp. 29–37. DOI: 10.1016/j.laa.2005.09.008
10. I. Gutman, N. N. M. de Abreu, C. T. M. Vinagre, A. S. Bonif´acio and S. Radenkovi´c (2008), Relation between energy and Laplacian energy, MATCH Communications in Mathematics and in Computer Chemistry, vol. 59, pp. 343-354.
11. K. M. Kathiresan, G. Marimuthu and M. Sivanandha Saraswathy (2010), Boundary domination in graphs, Kragujevac Journal of Mathematics, vol.33, pp.63–70.
12. J. Liu and B. Liu (2009), On relation between energy and Laplacian energy, MATCH Communications in Mathematics and in Computer Chemistry, vol. 61, pp. 403-406.
13. I. ˇZ. Milovanovi´c, E. I. Milovanovi´c and A. Zaki´c (2014), A short note on graph energy, MATCH Communications in Mathematics and in Computer Chemistry, vol. 72(1), pp. 179–182.
14. Mohammed Alatif and Puttaswamy (2016), Laplacian minimum boundary dominating energy of graphs, Asia Pacific Journal of Mathematics, vol. 3, issue no. 2, pp. 99-113, ISSN 2357-2205.
15. A. R. Nagalakshmi, A. S. Shrikanth, G. K. Kalavathi and K. S. Sreekeshava (2024), The Degree Energy of a Graph, MDPI, Σ Mathematics, vol. 12, page 2699. https://doi.org/10.3390/math12172699
16. K. Nataraj, Puttaswamy and S. Purushothama (2024), The Laplacian Minimum Pendant Dominating Energy of a Graph, Tuijin Jishu/Journal of Propulsion Technology, ISSN: 1001-4055, vol. 45, No. 3, pp. 3140-3150.
17. S. R. Nayaka, Puttaswamy and S. Purushothama (2020), Pendant Domination in Graphs, Journal of Combinatorial Mathematics and Combinatorial Computing, Vol. 112, pp. 219-230
18. S. R. Nayaka, Puttaswamy and S. Purushothama (2017), Complementary Pendant Domination in Graphs, International Journal of Pure and Applied Mathematics, Vol. 113, No. 11, pp. 179-187.
19. S. R. Nayaka, Puttaswamy and S. Purushothama (2017), Pendant Domination in Some Generalized Graphs, International Journal of Scientific Engineering and Science, vol. 1, No. 7, pp. 13-15.
20. S. R. Nayaka, Puttaswamy and S. Purushothama (2019), Minimum Pendant Dominating energy of a Graph, AIJRSTEM, ISSN (Print): 2328-3491, ISSN (Online): 2328-3580, ISSN (CD-ROM), pp. 2328-3629.
21. M. R. Rajesh Kanna, B. N. Dharmendra and G. Sridhara (2013), Minimum dominating energy of a graph, International Journal of Pure and Applied Mathematics, vol. 85(4), pp. 707-718.
22. M. R. Rajesh Kanna, B. N. Dharmendra and G. Sridhara (2013), Laplacian Minimum Dominating Energy of a Graph, International Journal of Pure and Applied Mathematics, vol. 89, No. 4, pp. 565-581.
23. D. StevanoviˆAtc, I. StankoviˆAtc and M. Milo¨EGseviˆAtc (2009) More on the relation between energy and Laplacian energy of graphs, MATCH Communications in Mathematical and in Computer Chemistry, vol. 61, pp. 395-401.
24. H. Wang and H. Hua (2008), Note on Laplacian energy of graphs, MATCH Communications in Mathematical and in Computer Chemistry, vol. 61, pp. 373-380.
25. B. Zhou (2009), New upper bounds for Laplacian energy, MATCH Communications in Mathematics and in Computer Chemistry, vol. 62, pp. 553-560.
26. B. Zhou (2008), On the sum of powers of the Laplacian eigenvalues of graphs, Linear Algebra and Its Applications, vol. 429(10), pp. 2239–2246. https://doi.org/10.1016/j.laa.2008.06.023
27. B. Zhou and I. Gutman (2007), On Laplacian energy of graphs, MATCH Communications in Mathematics and in Computer Chemistry, vol. 57, pp. 211-220.
Published
2026-03-27
Section
Research Articles
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