scholarly journals Sharp Upper and Lower Bounds of VDB Topological Indices of Digraphs

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1903
Author(s):  
Juan Monsalve ◽  
Juan Rada
Keyword(s):  
Lower Bounds ◽  
Topological Index ◽  
General Setting ◽  
Upper Bounds ◽  
Topological Indices ◽  
Vertex Degree ◽  
Upper And Lower Bounds ◽  
Lower And Upper Bounds ◽  
Extremal Values

A vertex-degree-based (VDB, for short) topological index φ induced by the numbers φij was recently defined for a digraph D, as φD=12∑uvφdu+dv−, where du+ denotes the out-degree of the vertex u,dv− denotes the in-degree of the vertex v, and the sum runs over the set of arcs uv of D. This definition generalizes the concept of a VDB topological index of a graph. In a general setting, we find sharp lower and upper bounds of a symmetric VDB topological index over Dn, the set of all digraphs with n non-isolated vertices. Applications to well-known topological indices are deduced. We also determine extremal values of symmetric VDB topological indices over OTn and OG, the set of oriented trees with n vertices, and the set of all orientations of a fixed graph G, respectively.

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 .

scholarly journals ORDERING CATACONDENSED HEXAGONAL SYSTEMS WITH RESPECT TO VDB TOPOLOGICAL INDICES

2017 ◽  
Vol 23 (1) ◽  
pp. 277-289
Author(s):  
Juan Rada
Keyword(s):  
Topological Index ◽  
Topological Indices ◽  
Vertex Degree ◽  
Extremal Values ◽  
Hexagonal Systems

In this paper we give a complete description of the ordering relations in the set of catacondensed hexagonal systems, with respect to a vertex-degree-based topological index. As a consequence, extremal values of vertex-degree-based topological indices in special subsets of the set of catacondensed hexagonal systems are computed.

scholarly journals Bounds for the variance of the busy period of the M/G/∞ queue

1984 ◽  
Vol 16 (4) ◽  
pp. 929-932 ◽  
Author(s):  
M. F. Ramalhoto
Keyword(s):  
Distribution Function ◽  
Lower Bounds ◽  
Service Time ◽  
Time Distribution ◽  
Busy Period ◽  
Random Variable ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Service Time Distribution ◽  
Lower And Upper Bounds

Some bounds for the variance of the busy period of an M/G/∞ queue are calculated as functions of parameters of the service-time distribution function. For any type of service-time distribution function, upper and lower bounds are evaluated in terms of the intensity of traffic and the coefficient of variation of the service time. Other lower and upper bounds are derived when the service time is a NBUE, DFR or IMRL random variable. The variance of the busy period is also related to the variance of the number of busy periods that are initiated in (0, t] by renewal arguments.

scholarly journals Bounds for the variance of the busy period of the M/G/∞ queue

1984 ◽  
Vol 16 (04) ◽  
pp. 929-932
Author(s):  
M. F. Ramalhoto
Keyword(s):  
Distribution Function ◽  
Lower Bounds ◽  
Service Time ◽  
Time Distribution ◽  
Busy Period ◽  
Random Variable ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Service Time Distribution ◽  
Lower And Upper Bounds

Some bounds for the variance of the busy period of an M/G/∞ queue are calculated as functions of parameters of the service-time distribution function. For any type of service-time distribution function, upper and lower bounds are evaluated in terms of the intensity of traffic and the coefficient of variation of the service time. Other lower and upper bounds are derived when the service time is a NBUE, DFR or IMRL random variable. The variance of the busy period is also related to the variance of the number of busy periods that are initiated in (0, t] by renewal arguments.

scholarly journals Spaces with high topological complexity

2014 ◽  
Vol 144 (4) ◽  
pp. 761-773
Author(s):  
Aleksandra Franc ◽  
Petar Pavešić
Keyword(s):  
Lower Bounds ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Topological Complexity ◽  
Lower And Upper Bounds ◽  
Cw Complex ◽  
The Difference

By a formula of Farber, the topological complexity TC(X) of a (p − 1)-connected m-dimensional CW-complex X is bounded above by (2m + 1)/p + 1. We show that the same result holds for the monoidal topological complexity TCM(X). In a previous paper we introduced various lower bounds for TCM(X), such as the nilpotency of the ring H*(X × X, Δ(X)), and the weak and stable (monoidal) topological complexity wTCM(X) and σTCM(X). In general, the difference between these upper and lower bounds can be arbitrarily large. In this paper we investigate spaces with topological complexity close to the maximal value given by Farber's formula. We show that in these cases the gap between the lower and upper bounds is narrow and TC(X) often coincides with the lower bounds.

scholarly journals Lower bounds for maximal cp-ranks of completely positive matrices and tensors

2020 ◽  
Vol 36 (36) ◽  
pp. 519-541
Author(s):  
Werner Schachinger
Keyword(s):  
Lower Bounds ◽  
Generating Function ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Lower And Upper Bounds ◽  
Completely Positive ◽  
Completely Positive Matrices ◽  
Positive Matrices ◽  
The Difference

Let $p_n$ denote the maximal cp-rank attained by completely positive $n\times n$ matrices. Only lower and upper bounds for $p_n$ are known, when $n\ge6$, but it is known that $p_n=\frac{n^2}2\big(1+o(1)\big)$, and the difference of the current best upper and lower bounds for $p_n$ is of order $\mathcal{O}\big(n^{3/2}\big)$. In this paper, that gap is reduced to $\mathcal{O}\big(n\log\log n\big)$. To achieve this result, a sequence of generalized ranks of a given matrix A has to be introduced. Properties of that sequence and its generating function are investigated. For suitable A, the $d$th term of that sequence is the cp-rank of some completely positive tensor of order $d$. This allows the derivation of asymptotically matching lower and upper bounds for the maximal cp-rank of completely positive tensors of order $d>2$ as well.

scholarly journals Bounds for eigenvalues of the Dirichlet problem for the logarithmic Laplacian

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Huyuan Chen ◽  
Laurent Véron
Keyword(s):  
Asymptotic Behavior ◽  
Fourier Transform ◽  
Dirichlet Problem ◽  
Lower Bounds ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Laplacian Operator ◽  
Lower And Upper Bounds

Abstract We provide bounds for the sequence of eigenvalues { λ i ⁢ ( Ω ) } i {\{\lambda_{i}(\Omega)\}_{i}} of the Dirichlet problem L Δ ⁢ u = λ ⁢ u ⁢  in  ⁢ Ω , u = 0 ⁢  in  ⁢ ℝ N ∖ Ω , L_{\Delta}u=\lambda u\text{ in }\Omega,\quad u=0\text{ in }\mathbb{R}^{N}% \setminus\Omega, where L Δ {L_{\Delta}} is the logarithmic Laplacian operator with Fourier transform symbol 2 ⁢ ln ⁡ | ζ | {2\ln\lvert\zeta\rvert} . The logarithmic Laplacian operator is not positively defined if the volume of the domain is large enough. In this article, we obtain the upper and lower bounds for the sum of the first k eigenvalues by extending the Li–Yau method and Kröger’s method, respectively. Moreover, we show the limit of the quotient of the sum of the first k eigenvalues by k ⁢ ln ⁡ k {k\ln k} is independent of the volume of the domain. Finally, we discuss the lower and upper bounds of the k-th principle eigenvalue, and the asymptotic behavior of the limit of eigenvalues.

scholarly journals Some New Bounds for Mathieu's Series

2007 ◽  
Vol 2007 ◽  
pp. 1-10 ◽  
Author(s):  
Abdolhossein Hoorfar ◽  
Feng Qi
Keyword(s):  
Lower Bounds ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Lower And Upper Bounds ◽  
Double Inequality

Two upper and lower bounds for Mathieu's series are established, which refine to a certain extent a sharp double inequality obtained by Alzer-Brenner-Ruehr in 1998. Moreover, the very closer lower and upper bounds forζ(3)are deduced.

scholarly journals Bounds and extremal graphs of second reformulated Zagreb index for graphs with cyclomatic number at most three

2021 ◽  
Vol 49 (1) ◽  
Author(s):  
Abhay Rajpoot ◽  
◽  
Lavanya Selvaganesh ◽  
Keyword(s):  
Lower Bounds ◽  
Topological Indices ◽  
Simple Approach ◽  
Vertex Degree ◽  
Upper And Lower Bounds ◽  
Extremal Graphs ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Cyclomatic Number ◽  
Tricyclic Graphs

Miliˇcevi´c et al., in 2004, introduced topological indices known as Reformulated Zagreb indices, where they modified Zagreb indices using the edge-degree instead of vertex degree. In this paper, we present a simple approach to find the upper and lower bounds of the second reformulated Zagreb index, EM2(G), by using six graph operations/transformations. We prove that these operations significantly alter the value of reformulated Zagreb index. We apply these transformations and identify those graphs with cyclomatic number at most 3, namely trees, unicyclic, bicyclic and tricyclic graphs, which attain the upper and lower bounds of second reformulated Zagreb index for graphs.

scholarly journals Multilevel Monte Carlo methods and lower–upper bounds in initial margin computations

2020 ◽  
Vol 26 (2) ◽  
pp. 131-161
Author(s):  
Florian Bourgey ◽  
Stefano De Marco ◽  
Emmanuel Gobet ◽  
Alexandre Zhou
Keyword(s):  
Monte Carlo ◽  
Lower Bounds ◽  
Asymptotic Optimality ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Independent Random Variables ◽  
Outer Function ◽  
Control Problems ◽  
Multilevel Monte Carlo ◽  
Primal Dual

AbstractThe multilevel Monte Carlo (MLMC) method developed by M. B. Giles [Multilevel Monte Carlo path simulation, Oper. Res. 56 2008, 3, 607–617] has a natural application to the evaluation of nested expectations {\mathbb{E}[g(\mathbb{E}[f(X,Y)|X])]}, where {f,g} are functions and {(X,Y)} a couple of independent random variables. Apart from the pricing of American-type derivatives, such computations arise in a large variety of risk valuations (VaR or CVaR of a portfolio, CVA), and in the assessment of margin costs for centrally cleared portfolios. In this work, we focus on the computation of initial margin. We analyze the properties of corresponding MLMC estimators, for which we provide results of asymptotic optimality; at the technical level, we have to deal with limited regularity of the outer function g (which might fail to be everywhere differentiable). Parallel to this, we investigate upper and lower bounds for nested expectations as above, in the spirit of primal-dual algorithms for stochastic control problems.

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