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 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 Exponential bounds and tails for additive random recursive sequences

2007 ◽  
Vol Vol. 9 no. 1 (Analysis of Algorithms) ◽  
Author(s):  
Ludger Rüschendorf ◽  
Eva-Maria Schopp
Keyword(s):  
Analysis Of Algorithms ◽  
Sufficient Conditions ◽  
Random Trees ◽  
Divide And Conquer ◽  
Recursive Sequences ◽  
Recursive Structures ◽  
International Audience ◽  
Exponential Bounds ◽  
The Moment

Analysis of Algorithms International audience Exponential bounds and tail estimates are derived for additive random recursive sequences, which typically arise as functionals of recursive structures, of random trees or in recursive algorithms. In particular they arise as parameters of divide and conquer type algorithms. We derive tail bounds from estimates of the Laplace transforms and of the moment sequences. For the proof we use some classical exponential bounds and some variants of the induction method. The paper generalizes results of Rösler (% \citeyearNPRoesler:91, % \citeyearNPRoesler:92) and % \citeNNeininger:05 on subgaussian tails to more general classes of additive random recursive sequences. It also gives sufficient conditions for tail bounds of the form \exp(-a t^p) which are based on a characterization of \citeNKasahara:78.

scholarly journals On the connectedness and diameter of a geometric Johnson graph

2013 ◽  
Vol Vol. 15 no. 3 (Combinatorics) ◽  
Author(s):  
Crevel Bautista-Santiago ◽  
Javier Cano ◽  
Ruy Fabila-Monroy ◽  
David Flores-Peñaloza ◽  
Hernàn González-Aguilar ◽  
...  
Keyword(s):  
Lower Bounds ◽  
General Position ◽  
Convex Set ◽  
Upper And Lower Bounds ◽  
Johnson Graph ◽  
International Audience

Combinatorics International audience Let P be a set of n points in general position in the plane. A subset I of P is called an island if there exists a convex set C such that I = P \C. In this paper we define the generalized island Johnson graph of P as the graph whose vertex consists of all islands of P of cardinality k, two of which are adjacent if their intersection consists of exactly l elements. We show that for large enough values of n, this graph is connected, and give upper and lower bounds on its diameter.

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.

Inequalities with application in economic risk analysis

1972 ◽  
Vol 9 (2) ◽  
pp. 441-444 ◽  
Author(s):  
Robert A. Agnew
Keyword(s):  
Risk Analysis ◽  
Lower Bounds ◽  
Generating Function ◽  
Random Variable ◽  
Moment Generating Function ◽  
Upper Bounds ◽  
Economic Risk ◽  
The Moment

Two sharp lower bounds for the expectation of a function of a non-negative random variable are obtained under rather weak hypotheses regarding the function, thus generalizing two sharp upper bounds obtained by Brook for the moment generating function. The application of these bounds to economic risk analysis is discussed.

SIGNATURE AND RELIABILITY OF CONDITIONAL THREE-DIMENSIONAL CONSECUTIVE-(s, s, s)-OUT-OF-(s, s, m):F SYSTEM

2014 ◽  
Vol 21 (02) ◽  
pp. 1450009 ◽  
Author(s):  
MADHURI G. KULKARNI ◽  
AKANKSHA S. KASHIKAR
Keyword(s):  
Lower Bounds ◽  
Three Dimensional ◽  
Upper And Lower Bounds ◽  
System Failure ◽  
Three Dimension ◽  
Two Dimensional ◽  
Math Model ◽  
The Moment ◽  
Appl Math ◽  
F System

A three-dimensional consecutive (r1, r2, r3)-out-of-(m1, m2, m3):F system was introduced by Akiba et al. [J. Qual. Mainten. Eng.11(3) (2005) 254–266]. They computed upper and lower bounds on the reliability of this system. Habib et al. [Appl. Math. Model.34 (2010) 531–538] introduced a conditional type of two-dimensional consecutive-(r, s)-out-of-(m, n):F system, where the number of failed components in the system at the moment of system failure cannot be more than 2rs. We extend this concept to three dimension and introduce a conditional three-dimensional consecutive (s, s, s)-out-of-(s, s, m):F system. It is an arrangement of ms2 components like a cuboid and it fails if it contains either a cube of failed components of size (s, s, s) or 2s3 failed components. We derive an expression for the signature of this system and also obtain reliability of this system using system signature.

Some limit theorems of runs to the continuous‐valued sequence

Kybernetes ◽  
2008 ◽  
Vol 37 (9/10) ◽  
pp. 1279-1286 ◽  
Author(s):  
Fan Aihua ◽  
Wang Zhongzhi ◽  
Ding Fangqing
Keyword(s):  
Limit Theorems ◽  
Random Sequence ◽  
Random Variables ◽  
Moment Generating Function ◽  
Upper And Lower Bounds ◽  
Content Type ◽  
Independent Case ◽  
The Moment ◽  
Martingale Convergence ◽  
Crucial Part

PurposeThe purpose of this paper is to give some limit theorems on ε‐neighborhood and ε‐increasing runs of continuous‐valued dependent random sequence. In the main result no assumptions are made concerning the random variables. As corollary a result on independent case is obtained.Design/methodology/approachThe crucial part of the proof is to construct a non‐negative supper‐martingale depending on a parameter by using the moment generating function, and then applying the Doob's martingale convergence theorem.FindingsThe upper and lower bounds of the deviations from the sums of arbitrary continuous‐valued random variables from the reference distributions are obtained.Research limitations/implicationsThe computation of asymptotic log‐likelihood ratio h(P|Q) is the main limitations, and it is difficult to obtain the rigorous bounds of the deviations.Practical implicationsA useful method to study the property for runs of dependent random sequence.Originality/valueThe new approach of study strong limit behavior for dependent random sequence.

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