Remark on upper bounds of Randi´c index of a graph

2016 ◽  
Vol 25 (1) ◽  
pp. 71-75
Author(s):  
I. Z. MILOVANOVIC ◽  
◽  
P. M. BEKAKOS ◽  
M. P. BEKAKOS ◽  
E. I. MILOVANOVIC ◽  
...  
Keyword(s):  
Upper Bound ◽  
Degree Sequence ◽  
Upper Bounds ◽  
Simple Graph ◽  
Vertex Degree ◽  
Randić Index ◽  
Randic Index ◽  
General Randić Index ◽  
General Randic Index

Let G = (V, E) be an undirected simple graph of order n with m edges without isolated vertices. Further, let d1 ≥ d2 ≥ · · · ≥ dn be vertex degree sequence of G. General Randic index of graph ´ G = (V, E) is defined by Rα = X (i,j)∈E (didj ) α, where α ∈ R − {0}. We consider the case when α = −1 and obtain upper bound for R−1.

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.

scholarly journals On the Randić Index of Corona Product Graphs

2011 ◽  
Vol 2011 ◽  
pp. 1-7
Author(s):  
Ismael G. Yero ◽  
Juan A. Rodríguez-Velázquez
Keyword(s):  
Degree Sequence ◽  
Upper Bounds ◽  
Vertex Degree ◽  
Regular Graphs ◽  
Randić Index ◽  
Lower And Upper Bounds ◽  
Randic Index ◽  
Product Graphs ◽  
Vertex Set ◽  
Corona Product

Let G be a graph with vertex set V=(v1,v2,…,vn). Let δ(vi) be the degree of the vertex vi∈V. If the vertices vi1,vi2,…,vih+1 form a path of length h≥1 in the graph G, then the hth order Randić index Rh of G is defined as the sum of the terms 1/δ(vi1)δ(vi2)⋯δ(vih+1) over all paths of length h contained (as subgraphs) in G. Lower and upper bounds for Rh, in terms of the vertex degree sequence of its factors, are obtained for corona product graphs. Moreover, closed formulas are obtained when the factors are regular graphs.

scholarly journals Some Upper Bounds on the First General Zagreb Index

2022 ◽  
Vol 2022 ◽  
pp. 1-4
Author(s):  
Muhammad Kamran Jamil ◽  
Aisha Javed ◽  
Ebenezer Bonyah ◽  
Iqra Zaman
Keyword(s):  
Minimum Degree ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Randić Index ◽  
Zeroth Order ◽  
Zagreb Index ◽  
Randic Index ◽  
Independent Number ◽  
General Randić Index ◽  
General Randic Index

The first general Zagreb index M γ G or zeroth-order general Randić index of a graph G is defined as M γ G = ∑ v ∈ V d v γ where γ is any nonzero real number, d v is the degree of the vertex v and γ = 2 gives the classical first Zagreb index. The researchers investigated some sharp upper and lower bounds on zeroth-order general Randić index (for γ < 0 ) in terms of connectivity, minimum degree, and independent number. In this paper, we put sharp upper bounds on the first general Zagreb index in terms of independent number, minimum degree, and connectivity for γ . Furthermore, extremal graphs are also investigated which attained the upper bounds.

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

General Randić index of unicyclic graphs with given diameter

2022 ◽  
Vol 306 ◽  
pp. 7-16
Author(s):  
Monther Rashed Alfuraidan ◽  
Kinkar Chandra Das ◽  
Tomáš Vetrík ◽  
Selvaraj Balachandran
Keyword(s):  
Randić Index ◽  
Unicyclic Graphs ◽  
Randic Index ◽  
General Randić Index ◽  
General Randic Index
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