scholarly journals Sharp bounds of the zeroth-order general Randić index of bicyclic graphs with given pendent vertices

2011 ◽  
Vol 159 (4) ◽  
pp. 240-245 ◽  
Author(s):  
Xiang-Feng Pan ◽  
Ning-Ning Lv
Keyword(s):  
Randić Index ◽  
Zeroth Order ◽  
Sharp Bounds ◽  
Randic Index ◽  
General Randić Index ◽  
General Randic Index ◽  
Bicyclic Graphs ◽  
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 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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