scholarly journals On the modified Randić index of trees, unicyclic graphs and bicyclic graphs

2012 ◽  
Vol 13 (2) ◽  
pp. 415 ◽  
Author(s):  
Jianping Li ◽  
Bo Zhou
Keyword(s):  
Randić Index ◽  
Unicyclic Graphs ◽  
Randic Index ◽  
Bicyclic Graphs

scholarly journals More results on extremum Randic indices of (molecular) trees

Filomat ◽  
2018 ◽  
Vol 32 (10) ◽  
pp. 3581-3590 ◽  
Author(s):  
Nor Husin ◽  
Roslan Hasni ◽  
Zhibin Du ◽  
Akbar Ali
Keyword(s):  
Randić Index ◽  
Unicyclic Graphs ◽  
Randic Index ◽  
The Third ◽  
Degree Of Vertex ◽  
Bicyclic Graphs ◽  
Molecular Trees

The Randic index R(G) of a graph G is the sum of the weights (dudv)-1/2 of all edges uv in G, where du denotes the degree of vertex u. Du and Zhou [On Randic indices of trees, unicyclic graphs, and bicyclic graphs, Int. J. Quantum Chem. 111 (2011), 2760-2770] determined the n-vertex trees with the third for n ? 7, the fourth for n ? 10, the fifth and the sixth for n ? 11 maximum Randic indices. Recently, Li et al. [The Randic indices of trees, unicyclic graphs and bicyclic graphs, Ars Comb. 127 (2016), 409-419] obtained the n-vertex trees with the seventh, the eighth, the ninth and the tenth for n ? 11 maximum Randic indices. In this paper, we correct the ordering for the Randic indices of trees obtained by Li et al., and characterize the trees with from the seventh to the sixteenth maximum Randic indices. The obtained extremal trees are molecular and thereby the obtained ordering also holds for molecular trees.

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

scholarly journals On a conjecture between Randic index and average distance of unicyclic graphs

Filomat ◽  
2014 ◽  
Vol 28 (4) ◽  
pp. 767-773
Author(s):  
Zhifu You ◽  
Bolian Liu
Keyword(s):  
Average Distance ◽  
Randić Index ◽  
Connected Graphs ◽  
Unicyclic Graphs ◽  
Randic Index

The Randic index R(G) of a graph G is defined as R(G) = ?uv?E (d(u)d(v))-1/2 where the summation goes over all edges of G. In 1988, Fajtlowicz proposed a conjecture: For all connected graphs G with average distance ad(G), then R(G) ? ad(G). In this paper, we prove that this conjecture is true for unicyclic graphs.

General Randić index of unicyclic graphs with given girth and diameter

2021 ◽  
Author(s):  
Tomáš Vetrík
Keyword(s):  
Lower Bounds ◽  
Randić Index ◽  
Zagreb Index ◽  
Unicyclic Graphs ◽  
Randic Index ◽  
General Randić Index ◽  
General Randic Index ◽  
Modified Zagreb Index

We study the general Randić index of a graph [Formula: see text], [Formula: see text], where [Formula: see text], [Formula: see text] is the edge set of [Formula: see text] and [Formula: see text] and [Formula: see text] are the degrees of vertices [Formula: see text] and [Formula: see text], respectively. For [Formula: see text], we present lower bounds on the general Randić index for unicyclic graphs of given diameter and girth, and unicyclic graphs of given diameter. Lower bounds on the classical Randić index and the second modified Zagreb index are corollaries of our results on the general Randić index.

scholarly journals Unicyclic graphs with extremal exponential Randić index

2021 ◽  
Vol 1 (3) ◽  
pp. 164-171
Author(s):  
Qian Lin ◽  
◽  
Yan Zhu
Keyword(s):  
Upper Bounds ◽  
Randić Index ◽  
Lower And Upper Bounds ◽  
Unicyclic Graphs ◽  
Randic Index ◽  
Degree Of A Vertex

<abstract><p>Recently the exponential Randić index $ {{\rm e}^{\chi}} $ was introduced. The exponential Randić index of a graph $ G $ is defined as the sum of the weights $ {{\rm e}^{{\frac {1}{\sqrt {d \left(u \right) d \left(v \right) }}}}} $ of all edges $ uv $ of $ G $, where $ d(u) $ denotes the degree of a vertex $ u $ in $ G $. In this paper, we give sharp lower and upper bounds on the exponential Randić index of unicyclic graphs.</p></abstract>

scholarly journals On the Minimum Variable Connectivity Index of Unicyclic Graphs with a Given Order

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Shamaila Yousaf ◽  
Akhlaq Ahmad Bhatti ◽  
Akbar Ali
Keyword(s):  
Real Number ◽  
Connectivity Index ◽  
Star Graph ◽  
Randić Index ◽  
Unicyclic Graphs ◽  
Randic Index ◽  
Degree Of A Vertex ◽  
Fixed Order ◽  
Negative Real Number

The variable connectivity index, introduced by the chemist Milan Randić in the first quarter of 1990s, for a graph G is defined as ∑vw∈EGdv+γdw+γ−1/2, where γ is a non-negative real number and dw is the degree of a vertex w in G. We call this index as the variable Randić index and denote it by Rvγ. In this paper, we show that the graph created from the star graph of order n by adding an edge has the minimum Rvγ value among all unicyclic graphs of a fixed order n, for every n≥4 and γ≥0.

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