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.

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 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.

Extremal graphs for the Randić index when minimum, maximum degrees and order of graphs are odd

Optimization ◽  
2014 ◽  
Vol 64 (9) ◽  
pp. 2021-2038 ◽  
Author(s):  
Tomica Divnić ◽  
Ljiljana Pavlović ◽  
Bolian Liu
Keyword(s):  
Randić Index ◽  
Extremal Graphs ◽  
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.

scholarly journals Extremal Trees with Respect to the Difference between Atom-Bond Connectivity Index and Randić Index

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1591
Author(s):  
Wan Nor Nabila Nadia Wan Zuki ◽  
Zhibin Du ◽  
Muhammad Kamran Jamil ◽  
Roslan Hasni
Keyword(s):  
Graph Theory ◽  
Connectivity Index ◽  
Randić Index ◽  
Extremal Graphs ◽  
Randic Index ◽  
Chemical Graph Theory ◽  
The Difference ◽  
Atom Bond ◽  
Chemical Trees

Let G be a simple, connected and undirected graph. The atom-bond connectivity index (ABC(G)) and Randić index (R(G)) are the two most well known topological indices. Recently, Ali and Du (2017) introduced the difference between atom-bond connectivity and Randić indices, denoted as ABC−R index. In this paper, we determine the fourth, the fifth and the sixth maximum chemical trees values of ABC−R for chemical trees, and characterize the corresponding extremal graphs. We also obtain an upper bound for ABC−R index of such trees with given number of pendant vertices. The role of symmetry has great importance in different areas of graph theory especially in chemical graph theory.

Minimal extremal graphs for addition of algebraic connectivity and independence number of connected graphs

Filomat ◽  
2017 ◽  
Vol 31 (18) ◽  
pp. 5545-5551
Author(s):  
Kinkar Das ◽  
Muhuo Liu
Keyword(s):  
Lower Bound ◽  
Independence Number ◽  
Laplacian Matrix ◽  
Algebraic Connectivity ◽  
Extremal Graphs ◽  
Laplacian Eigenvalue ◽  
Connected Graphs ◽  
Minimum Value ◽  
Vertex Degrees ◽  
Simple Connected Graph

Let G = (V,E) be a simple connected graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then the Laplacian matrix of graph G is L(G) = D(G)-A(G). Let a(G) and ?(G), respectively, be the second smallest Laplacian eigenvalue and the independence number of graph G. In this paper, we characterize the extremal graph with second minimum value for addition of algebraic connectivity and independence number among all connected graphs with n ? 6 vertices (Actually, we can determine the p-th minimum value of a(G)+ ?(G) under certain condition when p is small). Moreover, we present a lower bound to the addition of algebraic connectivity and radius of connected graphs.

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