Zeroth-order general Randić index of trees with given distance k-domination number

Tomas Vetrik, Mesfin Masre, Selvaraj Balachandran

Abstract


The zeroth-order general Randić index of a graph G is defined as Ra(G)=∑v ∈ V(G)dGa(v), where a ∈ ℝ, V(G) is the vertex set of G and dG(v) is the degree of a vertex v in G. We obtain bounds on the zeroth-order general Randić index for trees of given order and distance k-domination number, where k ≥ 1. Lower bounds are given for 0 < a < 1 and upper bounds are given for a < 0 and a > 1. All the extremal graphs are presented which means that our bounds are the best possible.

Keywords


zeroth-order general Randić index; tree; distance k-domination number

Full Text:

PDF

DOI: http://dx.doi.org/10.5614/ejgta.2022.10.1.17

References

H. Ahmed, A.A. Bhatti and A. Ali, Zeroth-order general Randic index of cactus graphs, AKCE Int. J. Graphs Comb. 16 (2019), 182–189.

https://www.sciencedirect.com/science/article/pii/S0972860017300774

A. Ali, M. Matejic, E. Milovanovi ´ c and I. Milovanovi ´ c, Some new upper bounds for the inverse sum indeg index of graphs, Electron. J. Graph Theory Appl. 8 (1) (2020), 59–70. https://www.ejgta.org/index.php/ejgta/article/view/618

S. Chen and H. Deng, Extremal (n, n + 1)-graphs with respected to zeroth-order general

Randic index, ´ J. Math. Chem. 42 (2007), 555–564.

https://link.springer.com/article/10.1007/s10910-006-9131-8

H. Deng, S. Balachandran, S. Elumalai and T. Mansour, Harary index of bipartite graphs, Electron. J. Graph Theory Appl. 7 (2) (2019), 365–372. https://www.ejgta.org/index.php/ejgta/article/view/710

M.K. Jamil, I. Tomescu, M. Imran and A. Javed, Some bounds on zeroth-order general Randic index, Mathematics 8 (2020), 1–12.

https://www.mdpi.com/2227-7390/8/1/98

S. Khalid and A. Ali, On the zeroth-order general Randic index, variable sum exdeg index and trees having vertices with prescribed degree, Discrete Math. Algorithms Appl. 10 (2018),

https://www.worldscientific.com/doi/10.1142/S1793830918500155

X. Li and H. Zhao, Trees with first three smallest and largest generalized topological indices, MATCH Commun. Math. Comput. Chem. 50 (2004), 57–62.

http://match.pmf.kg.ac.rs/electronic versions/Match50/match50 57-62.pdf

P. Milosevi ˇ c, I. Milovanovi ´ c, E. Milovanovi ´ c and M. Mateji ´ c, Some inequalities for general zeroth-order Randic index, ´ Filomat 33 (2019), 5249–5258. https://www.pmf.ni.ac.rs/filomat-content/2019/33-16/33-16-19-10987.pdf

L. Pei and X. Pan, Extremal values on Zagreb indices of trees with given distance kdomination number, J. Inequal. Appl. 16 (2018), 1–17. https://link.springer.com/article/10.1186/s13660-017-1597-3

I. Tomescu, On the general sum-connectivity index of connected graphs with given order and girth, Electron. J. Graph Theory Appl. 4 (1) (2016), 1–7. https://ejgta.org/index.php/ejgta/article/view/173/0

I. Tomescu and M.K. Jamil, Maximum general sum-connectivity index for trees with given independence number, MATCH Commun. Math. Comput. Chem. 72 (2014), 715–722. http://match.pmf.kg.ac.rs/electronic versions/Match72/n3/match72n3 715-722.pdf

T. Vetr´ık and B. Balachandran, Zeroth-order general Rand´ıc Index of trees, ´ Bol. Soc. Parana. Mat. 40 (2022), published online.

https://periodicos.uem.br/ojs/index.php/BSocParanMat/article/view/45062

T. Vetr´ık and M. Masre, General eccentric connectivity index of trees and unicyclic graphs, Discrete Appl. Math., 284 (2020), 301–315.

https://www.sciencedirect.com/science/article/abs/pii/S0166218X20301505

S. Yamaguchi, Zeroth-order general Randic index of trees with given order and distance conditions, MATCH Commun. Math. Comput. Chem. 62 (2009), 171–175. http://match.pmf.kg.ac.rs/electronic versions/Match62/n1/match62n1 171-175.pdf


Refbacks

  • There are currently no refbacks.


ISSN: 2338-2287

Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

View EJGTA Stats