scholarly journals Wiener-type invariants of some graph operations

Filomat ◽  
2009 ◽  
Vol 23 (3) ◽  
pp. 103-113 ◽  
Author(s):  
S. Hossein-Zadeh ◽  
A. Hamzeh ◽  
A.R. Ashrafi
Keyword(s):  
Real Number ◽  
Lower Bounds ◽  
Wiener Index ◽  
Upper And Lower Bounds ◽  
Harary Index ◽  
Graph Operations ◽  
Type Invariant ◽  
Wiener Type

Let d(G, k) be the number of pairs of vertices of a graph G that are at distance k, ? a real number, and W?(G) =?k?1 d(G, k)k?. W?(G) is called the Wiener-type invariant of G associated to real number ?. In this paper, the Wiener-type invariants of some graph operations are computed. As immediate consequences, the formulae for reciprocal Wiener index, Harary index, hyper- Wiener index and Tratch-Stankevich-Zefirov index are calculated. Some upper and lower bounds are also presented.

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

Bounds on hyper-Wiener index of graphs

2017 ◽  
Vol 10 (03) ◽  
pp. 1750057
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Sadegh Rahimi
Keyword(s):  
Lower Bounds ◽  
Organic Molecules ◽  
Wiener Index ◽  
Upper And Lower Bounds ◽  
Pharmacological Properties ◽  
Graph Theoretic ◽  
Acyclic Graphs ◽  
Wiener Number ◽  
Number Formula ◽  
Cyclic Graphs

The Wiener number [Formula: see text] of a graph [Formula: see text] was introduced by Harold Wiener in connection with the modeling of various physic-chemical, biological and pharmacological properties of organic molecules in chemistry. Milan Randić introduced a modification of the Wiener index for trees (acyclic graphs), and it is known as the hyper-Wiener index. Then Klein et al. generalized Randić’s definition for all connected (cyclic) graphs, as a generalization of the Wiener index, denoted by [Formula: see text] and defined as [Formula: see text]. In this paper, we establish some upper and lower bounds for [Formula: see text], in terms of other graph-theoretic parameters. Moreover, we compute hyper-Wiener number of some classes of graphs.

scholarly journals Wiener–Hosoya Matrix of Connected Graphs

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 359
Author(s):  
Hassan Ibrahim ◽  
Reza Sharafdini ◽  
Tamás Réti ◽  
Abolape Akwu
Keyword(s):  
Lower Bounds ◽  
Molecular Graph ◽  
Wiener Index ◽  
Vertex Degree ◽  
Upper And Lower Bounds ◽  
Connected Graphs ◽  
Matrix Description ◽  
Vertex Set

Let G be a connected (molecular) graph with the vertex set V(G)={v1,⋯,vn}, and let di and σi denote, respectively, the vertex degree and the transmission of vi, for 1≤i≤n. In this paper, we aim to provide a new matrix description of the celebrated Wiener index. In fact, we introduce the Wiener–Hosoya matrix of G, which is defined as the n×n matrix whose (i,j)-entry is equal to σi2di+σj2dj if vi and vj are adjacent and 0 otherwise. Some properties, including upper and lower bounds for the eigenvalues of the Wiener–Hosoya matrix are obtained and the extremal cases are described. Further, we introduce the energy of this matrix.

scholarly journals Hyper-Wiener indices of polyphenyl chains and polyphenyl spiders

2019 ◽  
Vol 17 (1) ◽  
pp. 668-676
Author(s):  
Tingzeng Wu ◽  
Huazhong Lü
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Wiener Index ◽  
Upper And Lower Bounds ◽  
Extremal Graphs ◽  
Recurrence Formulae

Abstract Let G be a connected graph and u and v two vertices of G. The hyper-Wiener index of graph G is $\begin{array}{} WW(G)=\frac{1}{2}\sum\limits_{u,v\in V(G)}(d_{G}(u,v)+d^{2}_{G}(u,v)) \end{array}$, where dG(u, v) is the distance between u and v. In this paper, we first give the recurrence formulae for computing the hyper-Wiener indices of polyphenyl chains and polyphenyl spiders. We then obtain the sharp upper and lower bounds for the hyper-Wiener index among polyphenyl chains and polyphenyl spiders, respectively. Moreover, the corresponding extremal graphs are determined.

scholarly journals Sharp bounds for a ratio of the q-gamma function in terms of the q-digamma function

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Faris Alzahrani ◽  
Ahmed Salem ◽  
Moustafa El-Shahed
Keyword(s):  
Real Number ◽  
Lower Bounds ◽  
Gamma Function ◽  
Open Problem ◽  
Upper And Lower Bounds ◽  
Digamma Function ◽  
Sharp Bounds ◽  
Gamma Functions ◽  
Numer Math

AbstractIn the present paper, we introduce sharp upper and lower bounds to the ratio of two q-gamma functions ${\Gamma }_{q}(x+1)/{\Gamma }_{q}(x+s)$ Γ q ( x + 1 ) / Γ q ( x + s ) for all real number s and $0< q\neq1$ 0 < q ≠ 1 in terms of the q-digamma function. Our results refine the results of Ismail and Muldoon (Internat. Ser. Numer. Math., vol. 119, pp. 309–323, 1994) and give the answer to the open problem posed by Alzer (Math. Nachr. 222(1):5–14, 2001). Also, for the classical gamma function, our results give a Kershaw inequality for all $0< s<1$ 0 < s < 1 when letting $q\to 1$ q → 1 and a new inequality for all $s>1$ s > 1 .

scholarly journals New Bounds for Topological Indices on Trees through Generalized Methods

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1097 ◽  
Author(s):  
Álvaro Martínez-Pérez ◽  
José M. Rodríguez
Keyword(s):  
Large Class ◽  
Lower Bounds ◽  
Maximum Degree ◽  
Chemical Compounds ◽  
Wiener Index ◽  
Topological Indices ◽  
Upper And Lower Bounds ◽  
Unified Approach ◽  
Wide Family ◽  
New Inequalities

Topological indices are useful for predicting the physicochemical behavior of chemical compounds. A main problem in this topic is finding good bounds for the indices, usually when some parameters of the graph are known. The aim of this paper is to use a unified approach in order to obtain several new inequalities for a wide family of topological indices restricted to trees and to characterize the corresponding extremal trees. The main results give upper and lower bounds for a large class of topological indices on trees, fixing or not the maximum degree. This class includes the first variable Zagreb, the Narumi–Katayama, the modified Narumi–Katayama and the Wiener index.

scholarly journals Sharp bounds on the zeroth-order general Randić index of trees in terms of domination number

2021 ◽  
Vol 7 (2) ◽  
pp. 2529-2542
Author(s):  
Chang Liu ◽  
◽  
Jianping Li
Keyword(s):  
Real Number ◽  
Lower Bounds ◽  
Domination Number ◽  
Upper And Lower Bounds ◽  
Randić Index ◽  
Extremal Graphs ◽  
Zeroth Order ◽  
Randic Index ◽  
General Randić Index ◽  
General Randic Index

<abstract><p>The zeroth-order general Randić index of graph $ G = (V_G, E_G) $, denoted by $ ^0R_{\alpha}(G) $, is the sum of items $ (d_{v})^{\alpha} $ over all vertices $ v\in V_G $, where $ \alpha $ is a pertinently chosen real number. In this paper, we obtain the sharp upper and lower bounds on $ ^0R_{\alpha} $ of trees with a given domination number $ \gamma $, for $ \alpha\in(-\infty, 0)\cup(1, \infty) $ and $ \alpha\in(0, 1) $, respectively. The corresponding extremal graphs of these bounds are also characterized.</p></abstract>

scholarly journals Tail Bounds for the Wiener Index of Random Trees

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Tämur Ali Khan ◽  
Ralph Neininger
Keyword(s):  
Lower Bounds ◽  
Random Search ◽  
Wiener Index ◽  
Moment Generating Function ◽  
Random Trees ◽  
Upper And Lower Bounds ◽  
Search Trees ◽  
Tail Probabilities ◽  
International Audience ◽  
The Moment

International audience Upper and lower bounds for the tail probabilities of the Wiener index of random binary search trees are given. For upper bounds the moment generating function of the vector of Wiener index and internal path length is estimated. For the lower bounds a tree class with sufficiently large probability and atypically large Wiener index is constructed. The methods are also applicable to related random search trees.

scholarly journals (β ,α)−Connectivity Index of Graphs

2017 ◽  
Vol 2 (1) ◽  
pp. 21-30 ◽  
Author(s):  
B. Basavanagoud ◽  
Veena R. Desai ◽  
Shreekant Patil
Keyword(s):  
Real Number ◽  
Lower Bounds ◽  
Connectivity Index ◽  
Upper And Lower Bounds ◽  
Line Graphs ◽  
Extremal Graphs ◽  
Acentric Factor ◽  
Wheel Graphs ◽  
Octane Isomers ◽  
Ladder Graphs

AbstractLet Eβ (G) be the set of paths of length β in a graph G. For an integer β ≥ 1 and a real number α, the (β,α)-connectivity index is defined as$$\begin{array}{} \displaystyle ^\beta\chi_\alpha(G)=\sum \limits_{v_1v_2 \cdot \cdot \cdot v_{\beta+1}\in E_\beta(G)}(d_{G}(v_1)d_{G}(v_2)...d_{G}(v_{\beta+1}))^{\alpha}. \end{array}$$The (2,1)-connectivity index shows good correlation with acentric factor of an octane isomers. In this paper, we compute the (2, α)-connectivity index of certain class of graphs, present the upper and lower bounds for (2, α)-connectivity index in terms of number of vertices, number of edges and minimum vertex degree and determine the extremal graphs which achieve the bounds. Further, we compute the (2, α)-connectivity index of line graphs of subdivision graphs of 2D-lattice, nanotube and nanotorus of TUC4C8[p,q], tadpole graphs, wheel graphs and ladder graphs.

scholarly journals Harmonic Index and Harmonic Polynomial on Graph Operations

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 456 ◽  
Author(s):  
Juan Hernández-Gómez ◽  
J. Méndez-Bermúdez ◽  
José Rodríguez ◽  
José M.  Sigarreta
Keyword(s):  
Lower Bounds ◽  
Convex Functions ◽  
Topological Index ◽  
Cartesian Product ◽  
Harmonic Polynomial ◽  
Upper And Lower Bounds ◽  
Lexicographic Product ◽  
Harmonic Index ◽  
Graph Operations ◽  
Corona Product

Some years ago, the harmonic polynomial was introduced to study the harmonic topological index. Here, using this polynomial, we obtain several properties of the harmonic index of many classical symmetric operations of graphs: Cartesian product, corona product, join, Cartesian sum and lexicographic product. Some upper and lower bounds for the harmonic indices of these operations of graphs, in terms of related indices, are derived from known bounds on the integral of a product on nonnegative convex functions. Besides, we provide an algorithm that computes the harmonic polynomial with complexity O ( n 2 ) .

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