scholarly journals Some Bounds for the Kirchhoff Index of Graphs

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Yujun Yang
Keyword(s):  
Connected Graph ◽  
Planar Graphs ◽  
Independence Number ◽  
Clique Number ◽  
Upper Bounds ◽  
Electrical Network ◽  
Lower And Upper Bounds ◽  
Kirchhoff Index ◽  
Resistance Distance ◽  
Fullerene Graphs

The resistance distance between two vertices of a connected graphGis defined as the effective resistance between them in the corresponding electrical network constructed fromGby replacing each edge ofGwith a unit resistor. The Kirchhoff index ofGis the sum of resistance distances between all pairs of vertices. In this paper, general bounds for the Kirchhoff index are given via the independence number and the clique number, respectively. Moreover, lower and upper bounds for the Kirchhoff index of planar graphs and fullerene graphs are investigated.

On the Kirchhoff Index of Graphs

2013 ◽  
Vol 68 (8-9) ◽  
pp. 531-538 ◽  
Author(s):  
Kinkar C. Das
Keyword(s):  
Connected Graph ◽  
Maximum Degree ◽  
Spanning Trees ◽  
Upper Bounds ◽  
Type Result ◽  
Lower And Upper Bounds ◽  
Kirchhoff Index ◽  
Laplacian Eigenvalues ◽  
Number Of Spanning Trees

Let G be a connected graph of order n with Laplacian eigenvalues μ1 ≥ μ2 ≥ ... ≥ μn-1 > mn = 0. The Kirchhoff index of G is defined as [xxx] In this paper. we give lower and upper bounds on Kf of graphs in terms on n, number of edges, maximum degree, and number of spanning trees. Moreover, we present lower and upper bounds on the Nordhaus-Gaddum-type result for the Kirchhoff index.

Resistance Distances and Kirchhoff Index in Generalised Join Graphs

2017 ◽  
Vol 72 (3) ◽  
pp. 207-215 ◽  
Author(s):  
Haiyan Chen
Keyword(s):  
Complete Graph ◽  
Connected Graph ◽  
Effective Resistance ◽  
Electrical Network ◽  
Kirchhoff Index ◽  
Resistance Distance ◽  
Join Graph ◽  
Explicit Formulae

AbstractThe resistance distance between any two vertices of a connected graph is defined as the effective resistance between them in the electrical network constructed from the graph by replacing each edge with a unit resistor. The Kirchhoff index of a graph is defined as the sum of all the resistance distances between any pair of vertices of the graph. Let G=H[G1, G2, …, Gk ] be the generalised join graph of G1, G2, …, Gk determined by H. In this paper, we first give formulae for resistance distances and Kirchhoff index of G in terms of parameters of ${G'_i}s$ and H. Then, we show that computing resistance distances and Kirchhoff index of G can be decomposed into simpler ones. Finally, we obtain explicit formulae for resistance distances and Kirchhoff index of G when ${G'_i}s$ and H take some special graphs, such as the complete graph, the path, and the cycle.

scholarly journals Resistance Distances in Linear Polyacene Graphs

2021 ◽  
Vol 8 ◽  
Author(s):  
Dayong Wang ◽  
Yujun Yang
Keyword(s):  
Connected Graph ◽  
Exact Expression ◽  
Effective Resistance ◽  
Electrical Network ◽  
Combinatorial Approach ◽  
Network Approach ◽  
Resistance Distance ◽  
Electric Network

The resistance distance between any two vertices of a connected graph is defined as the net effective resistance between them in the electrical network constructed from the graph by replacing each edge with a unit resistor. In this article, using electric network approach and combinatorial approach, we derive exact expression for resistance distances between any two vertices of polyacene graphs.

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.

Degree Kirchhoff Index of Bicyclic Graphs

2017 ◽  
Vol 60 (1) ◽  
pp. 197-205 ◽  
Author(s):  
Zikai Tang ◽  
Hanyuan Deng
Keyword(s):  
Connected Graph ◽  
Minimum Degree ◽  
Kirchhoff Index ◽  
Resistance Distance ◽  
Degree Of Vertex ◽  
Vertex Set ◽  
Maximum And Minimum Degree ◽  
Bicyclic Graphs

AbstractLet G be a connected graph with vertex set V(G).The degree Kirchhoò index of G is defined as S'(G) = Σ{u,v}⊆V(G) d(u)d(v)R(u, v), where d(u) is the degree of vertex u, and R(u, v) denotes the resistance distance between vertices u and v. In this paper, we characterize the graphs having maximum and minimum degree Kirchhoò index among all n-vertex bicyclic graphs with exactly two cycles.

scholarly journals New Bounds for Distance Energy of A Graph

2020 ◽  
Vol 26 (2) ◽  
pp. 213-223
Author(s):  
G. Sridhara ◽  
Rajesh Kanna ◽  
H.L. Parashivamurthy
Keyword(s):  
Connected Graph ◽  
Molecular Descriptor ◽  
Chemical Properties ◽  
Distance Matrix ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Physico Chemical ◽  
The Absolute

For any connected graph G, the distance energy, E_D(G) is defined as the sum of the absolute eigenvalues of its distance matrix.  Distance energy was introduced by Indulal et al in the year 2008. It has significant importance in QSPR analysis of molecular descriptor to study  their physico-chemical properties. Our interest in this article is to establish new lower and upper bounds for distance energy.

scholarly journals Some Extremal Graphs with Respect to Sombor Index

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1202
Author(s):  
Kinkar Chandra Das ◽  
Yilun Shang
Keyword(s):  
Molecular Structure ◽  
Chromatic Number ◽  
Clique Number ◽  
Upper Bounds ◽  
Extremal Graphs ◽  
Lower And Upper Bounds ◽  
Graph Parameters ◽  
Structure Descriptor

Let G be a graph with set of vertices V(G)(|V(G)|=n) and edge set E(G). Very recently, a new degree-based molecular structure descriptor, called Sombor index is denoted by SO(G) and is defined as SO=SO(G)=∑vivj∈E(G)dG(vi)2+dG(vj)2, where dG(vi) is the degree of the vertex vi in G. In this paper we present some lower and upper bounds on the Sombor index of graph G in terms of graph parameters (clique number, chromatic number, number of pendant vertices, etc.) and characterize the extremal graphs.

scholarly journals The general Albertson irregularity index of graphs

2021 ◽  
Vol 7 (1) ◽  
pp. 25-38
Author(s):  
Zhen Lin ◽  
◽  
Ting Zhou ◽  
Xiaojing Wang ◽  
Lianying Miao ◽  
...  
Keyword(s):  
Real Number ◽  
Connected Graph ◽  
Positive Real Number ◽  
Upper Bounds ◽  
Calculation Formula ◽  
Lower And Upper Bounds ◽  
Positive Real ◽  
Irregularity Index ◽  
Degree Of Vertex ◽  
Extremal Value

<abstract><p>We introduce the general Albertson irregularity index of a connected graph $ G $ and define it as $ A_{p}(G) = (\sum_{uv\in E(G)}|d(u)-d(v)|^p)^{\frac{1}{p}} $, where $ p $ is a positive real number and $ d(v) $ is the degree of the vertex $ v $ in $ G $. The new index is not only generalization of the well-known Albertson irregularity index and $ \sigma $-index, but also it is the Minkowski norm of the degree of vertex. We present lower and upper bounds on the general Albertson irregularity index. In addition, we study the extremal value on the general Albertson irregularity index for trees of given order. Finally, we give the calculation formula of the general Albertson index of generalized Bethe trees and Kragujevac trees.</p></abstract>

scholarly journals Lower and Upper Bounds for Hyper-Zagreb Index of Graphs

2020 ◽  
pp. 1401-1406
Author(s):  
G. H. SHIRDEL ◽  
H. REZAPOUR ◽  
R. NASIRI
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Chemical Compounds ◽  
Upper Bounds ◽  
Topological Indices ◽  
Upper And Lower Bounds ◽  
Lower And Upper Bounds ◽  
Zagreb Index ◽  
Simple Connected Graph

The topological indices are functions on the graph that do not depend on the labeling of their vertices. They are used by chemists for studying the properties of chemical compounds.  Let  be a simple connected graph. The Hyper-Zagreb index of the graph ,  is defined as  ,where  and  are the degrees of vertex  and , respectively. In this paper, we study the Hyper-Zagreb index and give upper and lower bounds for .

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