The general Randić index of trees with given number of pendent vertices

2017 ◽  
Vol 302 ◽  
pp. 111-121 ◽  
Author(s):  
Qing Cui ◽  
Lingping Zhong
Keyword(s):  
Randić Index ◽  
Randic Index ◽  
General Randić Index ◽  
General Randic Index ◽  
Pendent Vertices

scholarly journals On the zeroth-order general Randić index, variable sum exdeg index and trees having vertices with prescribed degree

2018 ◽  
Vol 10 (02) ◽  
pp. 1850015 ◽  
Author(s):  
Sohaib Khalid ◽  
Akbar Ali
Keyword(s):  
Real Number ◽  
Fixed Number ◽  
Randić Index ◽  
Zeroth Order ◽  
Positive Real ◽  
Randic Index ◽  
General Randić Index ◽  
General Randic Index ◽  
Pendent Vertices ◽  
Number Of Segments

The zeroth-order general Randić index (usually denoted by [Formula: see text]) and variable sum exdeg index (denoted by [Formula: see text]) of a graph [Formula: see text] are defined as [Formula: see text] and [Formula: see text], respectively, where [Formula: see text] is degree of the vertex [Formula: see text], [Formula: see text] is a positive real number different from 1 and [Formula: see text] is a real number other than [Formula: see text] and [Formula: see text]. A segment of a tree is a path [Formula: see text], whose terminal vertices are branching or/and pendent, and all non-terminal vertices (if exist) of [Formula: see text] have degree 2. For [Formula: see text], let [Formula: see text], [Formula: see text], [Formula: see text] be the collections of all [Formula: see text]-vertex trees having [Formula: see text] pendent vertices, [Formula: see text] segments, [Formula: see text] branching vertices, respectively. In this paper, all the trees with extremum (maximum and minimum) zeroth-order general Randić index and variable sum exdeg index are determined from the collections [Formula: see text], [Formula: see text], [Formula: see text]. The obtained extremal trees for the collection [Formula: see text] are also extremal trees for the collection of all [Formula: see text]-vertex trees having fixed number of vertices with degree 2 (because the number of segments of a tree [Formula: see text] can be determined from the number of vertices of [Formula: see text] having degree 2 and vice versa).

scholarly journals Bounds on General Randić Index for F-Sum Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-17
Author(s):  
Xu Li ◽  
Maqsood Ahmad ◽  
Muhammad Javaid ◽  
Muhammad Saeed ◽  
Jia-Bao Liu
Keyword(s):  
Molecular Graph ◽  
Randić Index ◽  
Lower And Upper Bounds ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
Randic Index ◽  
Structure Activity ◽  
General Randić Index ◽  
General Randic Index ◽  
The First Zagreb Index

A topological invariant is a numerical parameter associated with molecular graph and plays an imperative role in the study and analysis of quantitative structure activity/property relationships (QSAR/QSPR). The correlation between the entire π-electron energy and the structure of a molecular graph was explored and understood by the first Zagreb index. Recently, Liu et al. (2019) calculated the first general Zagreb index of the F-sum graphs. In the same paper, they also proposed the open problem to compute the general Randić index RαΓ=∑uv∈EΓdΓu×dΓvα of the F-sum graphs, where α∈R and dΓu denote the valency of the vertex u in the molecular graph Γ. Aim of this paper is to compute the lower and upper bounds of the general Randić index for the F-sum graphs when α∈N. We present numerous examples to support and check the reliability as well as validity of our bounds. Furthermore, the results acquired are the generalization of the results offered by Deng et al. (2016), who studied the general Randić index for exactly α=1.

scholarly journals Connected (n,m)-graphs with minimum and maximum zeroth-order general Randić index

2007 ◽  
Vol 155 (8) ◽  
pp. 1044-1054 ◽  
Author(s):  
Yumei Hu ◽  
Xueliang Li ◽  
Yongtang Shi ◽  
Tianyi Xu
Keyword(s):  
Randić Index ◽  
Zeroth Order ◽  
Randic Index ◽  
General Randić Index ◽  
General Randic Index

scholarly journals Some Bounds on Zeroth-Order General Randić Index

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 98 ◽  
Author(s):  
Muhammad Kamran Jamil ◽  
Ioan Tomescu ◽  
Muhammad Imran ◽  
Aisha Javed
Keyword(s):  
Clique Number ◽  
Upper Bounds ◽  
Randić Index ◽  
Extremal Graphs ◽  
Lower And Upper Bounds ◽  
Zeroth Order ◽  
Randic Index ◽  
General Randić Index ◽  
Inverse Degree ◽  
General Randic Index

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.

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