scholarly journals Reciprocal product-degree distance of graphs

Filomat ◽  
2016 ◽  
Vol 30 (8) ◽  
pp. 2217-2231
Author(s):  
Guifu Su ◽  
Liming Xiong ◽  
Ivan Gutman ◽  
Lan Xu
Keyword(s):  
Connected Graph ◽  
Upper Bounds ◽  
Graph Invariant ◽  
Lower And Upper Bounds ◽  
Harary Index ◽  
Underlying Graph ◽  
Degree Distance ◽  
Degree Modification ◽  
Type Relation

We investigate a new graph invariant named reciprocal product-degree distance, defined as: RDD* = ?{u,v}?V(G)u?v deg(u)?deg(v)/dist(u,v) where deg(v) is the degree of the vertex v, and dist(u,v) is the distance between the vertices u and v in the underlying graph. RDD* is a product-degree modification of the Harary index. We determine the connected graph of given order with maximum RDD*-value, and establish lower and upper bounds for RDD*. Also a Nordhaus-Gaddum-type relation for RDD* is obtained.

scholarly journals Harary index of the k-th power of a graph

2013 ◽  
Vol 7 (1) ◽  
pp. 94-105 ◽  
Author(s):  
Guifu Su ◽  
Liming Xiong ◽  
Ivan Gutman
Keyword(s):  
Type Inequality ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Harary Index ◽  
Underlying Graph

The k-th power of a graph G, denoted by Gk, is a graph with the same set of vertices as G, such that two vertices are adjacent in Gk if and only if their distance in G is at most k. The Harary index H is the sum of the reciprocal distances of all pairs of vertices of the underlying graph. Lower and upper bounds on H(Gk) are obtained. A Nordhaus-Gaddum type inequality for H(Gk) is also established.

scholarly journals On the reformulated reciprocal degree distance of graphs

2016 ◽  
Vol 25 (2) ◽  
pp. 205-213
Author(s):  
K. PATTABIRAMAN ◽  
◽  
M. VIJAYARAGAVAN ◽  
Keyword(s):  
Connected Graph ◽  
Cartesian Product ◽  
Vertex Degree ◽  
Symmetric Difference ◽  
Weighted Sum ◽  
Graph Invariant ◽  
Harary Index ◽  
Degree Distance

The reciprocal degree distance (RDD), defined for a connected graph G as vertex-degree-weighted sum of the reciprocal distances, that is, RDD(G) = P u,v∈V (G) (d(u)+d(v)) dG(u,v) . The new graph invariant named reformulated reciprocal degree distance is defined for a connected graph G as Rt(G) = P u,v∈V (G) (d(u)+d(v)) dG(u,v)+t , t ≥ 0. The reformulated reciprocal degree distance is a weight version of the t-Harary index, that is, Ht(G) = P u,v∈V (G) 1 dG(u,v)+t , t ≥ 0. In this paper, the reformulated reciprocal degree distance and reciprocal degree distance of disjunction, symmetric difference, Cartesian product of two graphs are obtained. Finally, we obtain the reformulated reciprocal degree distance and reciprocal degree distance of double a graph.

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.

Upper bounds on product degree distance of F-sum of graphs

2019 ◽  
Vol 11 (04) ◽  
pp. 1950045
Author(s):  
K. Pattabiraman ◽  
Manzoor Ahmad Bhat
Keyword(s):  
Connected Graph ◽  
Upper Bounds ◽  
Degree Of A Vertex ◽  
Degree Distance ◽  
Appl Math

The product degree distance of a connected graph [Formula: see text] is defined as [Formula: see text] where [Formula: see text] is the degree of a vertex [Formula: see text] and [Formula: see text] is the distance between the vertices [Formula: see text] and [Formula: see text] in [Formula: see text] In this paper, we obtain two upper bounds for product degree distance of [Formula: see text]-sums of graphs which is defined by Eliasi [Four new sums of graphs and their Wiener indices, Discr. Appl. Math. 157 (2009) 794–803.]

scholarly journals ESTRADA INDEX OF GENERAL WEIGHTED GRAPHS

2012 ◽  
Vol 88 (1) ◽  
pp. 106-112 ◽  
Author(s):  
YILUN SHANG
Keyword(s):  
Weighted Graph ◽  
Upper Bounds ◽  
Weighted Graphs ◽  
Graph Invariant ◽  
Lower And Upper Bounds ◽  
Spectral Moments ◽  
Estrada Index ◽  
Low Order

AbstractLet $G$ be a general weighted graph (with possible self-loops) on $n$ vertices and $\lambda _1,\lambda _2,\ldots ,\lambda _n$ be its eigenvalues. The Estrada index of $G$ is a graph invariant defined as $EE=\sum _{i=1}^ne^{\lambda _i}$. We present a generic expression for $EE$ based on weights of short closed walks in $G$. We establish lower and upper bounds for $EE$in terms of low-order spectral moments involving the weights of closed walks. A concrete example of calculation is provided.

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.

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.

scholarly journals Reformulated Reciprocal Degree Distance and Reciprocal Degree Distance of the Complement of the Mycielskian Graph and Generalized Mycielskian

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Feifei Zhao ◽  
Hong Bian ◽  
Haizheng Yu ◽  
Min Liu
Keyword(s):  
Connected Graph ◽  
Harary Index ◽  
Degree Distance

The reformulated reciprocal degree distance is defined for a connected graph G as R¯t(G)=(1/2)∑u,υ∈VG((dG(u)+dG(υ))/(dG(u,υ)+t)),t≥0, which can be viewed as a weight version of the t-Harary index; that is, H¯t(G)=(1/2)∑u,υ∈VG(1/(dG(u,υ)+t)),t≥0. In this paper, we present the reciprocal degree distance index of the complement of Mycielskian graph and generalize the corresponding results to the generalized Mycielskian graph.

scholarly journals On the Harary index of cacti

Filomat ◽  
2014 ◽  
Vol 28 (3) ◽  
pp. 493-507 ◽  
Author(s):  
Zhongxun Zhu ◽  
Ting Tao ◽  
Jing Yu ◽  
Liansheng Tan
Keyword(s):  
Connected Graph ◽  
Perfect Matching ◽  
Upper Bounds ◽  
Common Vertex ◽  
Harary Index

The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. A connected graph G is a cactus if any two of its cycles have at most one common vertex. Let G(n, r) be the set of cacti of order n and with r cycles, ?(2n,r) the set of cacti of order 2n with a perfect matching and r cycles. In this paper, we give the sharp upper bounds of the Harary index of cacti among G (n,r) and ?(2n, r), respectively, and characterize the corresponding extremal cactus.

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>

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