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.

scholarly journals On the Number of Spanning Trees of Graphs

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Ş. Burcu Bozkurt ◽  
Durmuş Bozkurt
Keyword(s):  
Spanning Trees ◽  
Vertex Degree ◽  
Randić Index ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
Connected Graphs ◽  
Randic Index ◽  
Number Of Spanning Trees

We establish some bounds for the number of spanning trees of connected graphs in terms of the number of vertices(n), the number of edges(m), maximum vertex degree(Δ1), minimum vertex degree(δ),…first Zagreb index(M1),and Randić index(R-1).

scholarly journals On the variation of the Randic index with given girth and leaves

Filomat ◽  
2014 ◽  
Vol 28 (9) ◽  
pp. 1849-1853 ◽  
Author(s):  
Jianxi Liu
Keyword(s):  
Randić Index ◽  
Extremal Graphs ◽  
Connected Graphs ◽  
Randic Index ◽  
Minimum Value ◽  
Degree Of A Vertex

The variation of Randic index R'(G) of a graph G is defined by R'(G) = ?uv 1/ max{du,dv}, where du is the degree of a vertex u in G and the summation extends over all edges uv of G. In this work, we characterize the extremal trees achieving the minimum value of R0 for trees with given number of vertices and leaves. Furthermore, we characterize the extremal graphs achieving the minimum value of R' for connected graphs with given number of vertices and girth.

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 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>

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.

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