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Open AccessArticle

# Some Bounds on Zeroth-Order General Randić Index

by Muhammad Kamran Jamil 1,2,*, Ioan Tomescu 3, Muhammad Imran 2,4 and Aisha Javed 5
1
Department of Mathematics, Riphah Institute of Computing and Applied Sciences, Riphah International University, P. O. Box 54600, Lahore, Pakistan
2
Department of Mathematical Sciences, United Arab Emirates University, P. O. Box 15551, Al Ain, UAE
3
Faculty of Mathematics and Computer Science, University of Bucharest, P. O. Box 050663, Bucharest, Romania
4
Department of Mathematics, School of Natural Sciences, NUST Islamabad, P. O. Box 24090, Islamabad, Pakistan
5
Abdus Salam School of Mathematical Sciences, GC University, P. O. Box 54600, Lahore, Pakistan
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(1), 98; https://doi.org/10.3390/math8010098
Received: 15 December 2019 / Revised: 2 January 2020 / Accepted: 2 January 2020 / Published: 7 January 2020
(This article belongs to the Special Issue Graph Theory at Work in Carbon Chemistry)
For a graph G without isolated vertices, the inverse degree of a graph G is defined as $I D ( G ) = ∑ u ∈ V ( G ) d ( u ) − 1$ where $d ( u )$ is the number of vertices adjacent to the vertex u in G. By replacing $− 1$ by any non-zero real number we obtain zeroth-order general Randić index, i.e., $0 R γ ( G ) = ∑ u ∈ V ( G ) d ( u ) γ$ , where $γ ∈ R − { 0 }$ . Xu et al. investigated some lower and upper bounds on $I D$ for a connected graph G in terms of connectivity, chromatic number, number of cut edges, and clique number. In this paper, we extend their results and investigate if the same results hold for $γ < 0$ . The corresponding extremal graphs have also been identified. View Full-Text
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MDPI and ACS Style

Jamil, M.K.; Tomescu, I.; Imran, M.; Javed, A. Some Bounds on Zeroth-Order General Randić Index. Mathematics 2020, 8, 98.