scholarly journals Further results on non-self-centrality measures of graphs

Filomat ◽  
2018 ◽  
Vol 32 (14) ◽  
pp. 5137-5148 ◽  
Author(s):  
Mahdieh Azari
Keyword(s):  
Structural Parameters ◽  
Cartesian Product ◽  
Upper And Lower Bounds ◽  
Centrality Measures ◽  
Eccentric Connectivity Index ◽  
The Third ◽  
Eccentricity Index ◽  
Strong Product ◽  
Special Cases ◽  
Special Case

For indicating the non-self-centrality extent of graphs, two new eccentricity-based measures namely third Zagreb eccentricity index E3(G) and non-self-centrality number N(G) of a connected graph G have recently been introduced as E3(G) = ?uv?E(G)|?G(u)-?G(v)| and N(G) = ? {u,v}?V(G) |?G(u)-?G(v)|, where ?G(u) denotes the eccentricity of a vertex u in G. In this paper, we find relation between the third Zagreb eccentricity index of graphs with some eccentricity-based invariants such as second Zagreb eccentricity index and second eccentric connectivity index. We also give sharp upper and lower bounds on the nonself-centrality number of graphs in terms of some structural parameters and relate it to two well-known eccentricity-based invariants namely total eccentricity and first Zagreb eccentricity index. Furthermore, we present exact expressions or sharp upper bounds on the third Zagreb eccentricity index and non-selfcentrality number of several graph operations such as join, disjunction, symmetric difference, lexicographic product, strong product, and generalized hierarchical product. The formulae for Cartesian product and rooted product as two important special cases of generalized hierarchical product and the formulae for corona product as a special case of rooted product are also given.

A novel graph invariant: The third leap Zagreb index under several graph operations

2019 ◽  
Vol 11 (05) ◽  
pp. 1950054 ◽  
Author(s):  
Durbar Maji ◽  
Ganesh Ghorai
Keyword(s):  
Tensor Product ◽  
Cartesian Product ◽  
Lexicographic Product ◽  
Graph Invariant ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
The Third ◽  
Graph Operations ◽  
Strong Product ◽  
Corona Product

The third leap Zagreb index of a graph [Formula: see text] is denoted as [Formula: see text] and is defined as [Formula: see text], where [Formula: see text] and [Formula: see text] are the 2-distance degree and the degree of the vertex [Formula: see text] in [Formula: see text], respectively. The first, second and third leap Zagreb indices were introduced by Naji et al. [A. M. Naji, N. D. Soner and I. Gutman, On leap Zagreb indices of graphs, Commun. Combin. Optim. 2(2) (2017) 99–117] in 2017. In this paper, the behavior of the third leap Zagreb index under several graph operations like the Cartesian product, Corona product, neighborhood Corona product, lexicographic product, strong product, tensor product, symmetric difference and disjunction of two graphs is studied.

scholarly journals ENERGY TRANSMISSION OVER BREAK WATER - A DESIGN CRITERION?

1980 ◽  
Vol 1 (17) ◽  
pp. 114
Author(s):  
P. Bade ◽  
H. Kaldenhoff
Keyword(s):  
Structural Parameters ◽  
Wave Climate ◽  
Design Criterion ◽  
Energy Transmission ◽  
Transmission Coefficients ◽  
Wave Parameters ◽  
Special Cases ◽  
Special Case ◽  
Break Water

The wave transmission from seaside to lee of break waters depends on structural parameters as well as on the initial wave climate. Transmission coefficients for special boundary: conditions are known, but information about wave parameters are often not available or they cover often only special cases. Again a case study, undertaken for the break water at the proposed deep water harbour at Scharhoern/Neuwerk, Germany, documented this lack. The results of the case study using conservative wave parameters did not answer the important questions of energy transmission sufficiently, thus we investigated the characteristics of the initial and transmitted wave spectra. A special case was treated initially and subsequently more general situations were examined. These tests were carried out using a physical model of a length scale of 1:10 under conditions described by Froude's law.

scholarly journals Parameterized Approximation Algorithms for Bidirected Steiner Network Problems

2021 ◽  
Vol 17 (2) ◽  
pp. 1-68
Author(s):  
Rajesh Chitnis ◽  
Andreas Emil Feldmann ◽  
Pasin Manurangsi
Keyword(s):  
Approximation Algorithm ◽  
Approximation Scheme ◽  
Upper And Lower Bounds ◽  
Important Special Case ◽  
Bidirected Graph ◽  
Negative Edge ◽  
Network Problems ◽  
Special Cases ◽  
Bidirected Graphs ◽  
Special Case

The D irected S teiner N etwork (DSN) problem takes as input a directed graph G =( V , E ) with non-negative edge-weights and a set D ⊆ V × V of k demand pairs. The aim is to compute the cheapest network N⊆ G for which there is an s\rightarrow t path for each ( s , t )∈ D. It is known that this problem is notoriously hard, as there is no k 1/4− o (1) -approximation algorithm under Gap-ETH, even when parametrizing the runtime by k [Dinur & Manurangsi, ITCS 2018]. In light of this, we systematically study several special cases of DSN and determine their parameterized approximability for the parameter k . For the bi -DSNP lanar problem, the aim is to compute a solution N⊆ G whose cost is at most that of an optimum planar solution in a bidirected graph G , i.e., for every edge uv of G the reverse edge vu exists and has the same weight. This problem is a generalization of several well-studied special cases. Our main result is that this problem admits a parameterized approximation scheme (PAS) for  k . We also prove that our result is tight in the sense that (a) the runtime of our PAS cannot be significantly improved, and (b) no PAS exists for any generalization of bi-DSNP lanar , under standard complexity assumptions. The techniques we use also imply a polynomial-sized approximate kernelization scheme (PSAKS). Additionally, we study several generalizations of bi -DSNP lanar and obtain upper and lower bounds on obtainable runtimes parameterized by  k . One important special case of DSN is the S trongly C onnected S teiner S ubgraph (SCSS) problem, for which the solution network N⊆ G needs to strongly connect a given set of k terminals. It has been observed before that for SCSS a parameterized 2-approximation exists for parameter  k [Chitnis et al., IPEC 2013]. We give a tight inapproximability result by showing that for k no parameterized (2 − ε)-approximation algorithm exists under Gap-ETH. Additionally, we show that when restricting the input of SCSS to bidirected graphs, the problem remains NP-hard but becomes FPT for k .

Three Perspectives on Centrality

2020 ◽  
pp. 333-351
Author(s):  
Stephen P. Borgatti ◽  
Martin G. Everett
Keyword(s):  
Centrality Measures ◽  
Graph Invariant ◽  
Propagation Process ◽  
Centrality Measure ◽  
The Third ◽  
First Arrival ◽  
The Difference

This chapter presents three different perspectives on centrality. In part, the motivation is definitional: what counts as a centrality measure and what doesn’t? But the primary purpose is to lay out ways that centrality measures are similar and dissimilar and point to appropriate ways of interpreting different measures. The first perspective the chapter considers is the “walk structure participation” perspective. In this perspective, centrality measures indicate the extent and manner in which a node participates in the walk structure of a graph. A typology is presented that distinguishes measures based on dimensions such as (1) what kinds of walks are considered (e.g., geodesics, paths, trails, or unrestricted walks) and (2) whether the number of walks is counted or the length of walks is assessed, or both. The second perspective the chapter presents is the “induced centrality” perspective, which views a node’s centrality as its contribution to a specific graph invariant—typically some measure of the cohesiveness of the network. Induced centralities are computed by calculating the graph invariant, removing the node in question, and recalculating the graph invariant. The difference is the node’s centrality. The third perspective is the “flow outcomes” perspective. Here the chapter views centralities as estimators of node outcomes in some kind of propagation process. Generic node outcomes include how often a bit of something propagating passes through a node and the time until first arrival of something flowing. The latter perspective leads us to consider the merits of developing custom measures for different research settings versus using off-the-shelf measures that were not necessarily designed for the current purpose.

Analytical Techniques for the Optimisation of Rocket Trajectories

1963 ◽  
Vol 14 (2) ◽  
pp. 105-124 ◽  
Author(s):  
Derek F. Lawden
Keyword(s):  
Gravitational Field ◽  
Artificial Satellite ◽  
Analytical Techniques ◽  
Present Position ◽  
Mayer Problem ◽  
Special Cases ◽  
Minimum Expenditure ◽  
Law Of Attraction ◽  
Special Case

SummaryThe development during the last two decades of analytical techniques for the solution of problems relating to the optimisation of rocket trajectories is outlined and the present position in this field of research is summarised. It is shown that the determination of optimal trajectories in a general gravitational field can be expressed as a Mayer problem from the calculus of variations. The known solution to such a problem is stated and applied, first to the special case of the launching of an artificial satellite into a circular orbit with minimum expenditure of propellant and, secondly, to the general astronautical problem of the economical transfer of a rocket between two terminals in a gravitational field. The special cases when the field is uniform and when it obeys an inverse square law of attraction to a point are then considered, and the paper concludes with some remarks concerning areas in which further investigations are necessary.

scholarly journals Water entry of an expanding wedge/plate with flow detachment

2016 ◽  
Vol 797 ◽  
pp. 322-344 ◽  
Author(s):  
Yuriy A. Semenov ◽  
Guo Xiong Wu
Keyword(s):  
Free Surface ◽  
General Solution ◽  
Water Entry ◽  
Hodograph Method ◽  
Fixed Length ◽  
Pressure Distributions ◽  
Initial Stage ◽  
Special Cases ◽  
Wedge Plate ◽  
Special Case

A general similarity solution for water-entry problems of a wedge with its inner angle fixed and its sides in expansion is obtained with flow detachment, in which the speed of expansion is a free parameter. The known solutions for a wedge of a fixed length at the initial stage of water entry without flow detachment and at the final stage corresponding to Helmholtz flow are obtained as two special cases, at some finite and zero expansion speeds, respectively. An expanding horizontal plate impacting a flat free surface is considered as the special case of the general solution for a wedge inner angle equal to ${\rm\pi}$. An initial impulse solution for a plate of a fixed length is obtained as the special case of the present formulation. The general solution is obtained in the form of integral equations using the integral hodograph method. The results are presented in terms of free-surface shapes, streamlines and pressure distributions.

scholarly journals The Steady Profile of an Axisymmetric Ice Sheet

1981 ◽  
Vol 27 (95) ◽  
pp. 25-37 ◽  
Author(s):  
I. R. Johnson
Keyword(s):  
Aspect Ratio ◽  
Power Law ◽  
Viscous Incompressible Fluid ◽  
Ice Sheet ◽  
Plane Flow ◽  
The Third ◽  
Basal Sliding ◽  
Steady Plane ◽  
Third Dimension ◽  
Special Case

AbstractSteady plane flow under gravity of an axisymmetric ice sheet resting on a horizontal rigid bed, subject to surface accumulation and ablation, basal drainage, and basal sliding is treated according to a power law between shear traction and velocity. The surface accumulation is taken to depend on height, and the drainage and sliding coefficient also depend on the height of overlying ice. The ice is described as a general non-linearly viscous incompressible fluid, and temperature variation through the ice sheet is neglected. Illustrations are presented for Glen’s power law (including the special case of a Newtonian fluid), and the polynomial law of Colbeck and Evans. The analysis follows that of Morland and Johnson (1980) where the analogous problem for an ice sheet deforming under plane flow was considered. Comparisons are made between the two models and it is found that the effect of the third dimension is to reduce (or leave unchanged) the aspect ratio for the cases considered, although no general formula can be obtained. This reduction is seen to depend on both the surface accumulation and the sliding law.

scholarly journals On Multivalent Analytic Functions Considered by a Multi-Arbitrary Differential Operator in a Complex Domain

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 315
Author(s):  
Najla M. Alarifi ◽  
Rabha W. Ibrahim
Keyword(s):  
Differential Operator ◽  
Analytic Functions ◽  
Integral Operators ◽  
Upper And Lower Bounds ◽  
Fractional Operator ◽  
Convolution Operators ◽  
Linear Convolution ◽  
Fractional Differential Operator ◽  
Fractional Differential ◽  
Special Case

(1) Background: There is an increasing amount of information in complex domains, which necessitates the development of various kinds of operators, such as differential, integral, and linear convolution operators. Few investigations of the fractional differential and integral operators of a complex variable have been undertaken. (2) Methods: In this effort, we aim to present a generalization of a class of analytic functions based on a complex fractional differential operator. This class is defined by utilizing the subordination and superordination theory. (3) Results: We illustrate different fractional inequalities of starlike and convex formulas. Moreover, we discuss the main conditions to obtain sandwich inequalities involving the fractional operator. (4) Conclusion: We indicate that the suggested class is a generalization of recent works and can be applied to discuss the upper and lower bounds of a special case of fractional differential equations.

scholarly journals On $ {L}(2,1) $-labelings of some products of oriented cycles

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Lucas Colucci ◽  
Ervin Győri
Keyword(s):  
Cartesian Product ◽  
Strong Product

<p style='text-indent:20px;'>We refine two results of Jiang, Shao and Vesel on the <inline-formula><tex-math id="M2">\begin{document}$ L(2,1) $\end{document}</tex-math></inline-formula>-labeling number <inline-formula><tex-math id="M3">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> of the Cartesian and the strong product of two oriented cycles. For the Cartesian product, we compute the exact value of <inline-formula><tex-math id="M4">\begin{document}$ \lambda(\overrightarrow{C_m} \square \overrightarrow{C_n}) $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M5">\begin{document}$ m $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ n \geq 40 $\end{document}</tex-math></inline-formula>; in the case of strong product, we either compute the exact value or establish a gap of size one for <inline-formula><tex-math id="M7">\begin{document}$ \lambda(\overrightarrow{C_m} \boxtimes \overrightarrow{C_n}) $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M8">\begin{document}$ m $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M9">\begin{document}$ n \geq 48 $\end{document}</tex-math></inline-formula>.</p>

scholarly journals On Reay's Relaxed Tverberg Conjecture and Generalizations of Conway's Thrackle Conjecture

10.37236/6516 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Megumi Asada ◽  
Ryan Chen ◽  
Florian Frick ◽  
Frederick Huang ◽  
Maxwell Polevy ◽  
...  
Keyword(s):  
Geometric Property ◽  
Convex Sets ◽  
Convex Hulls ◽  
Open Problems ◽  
Higher Dimensional ◽  
Special Cases ◽  
Minimum Number ◽  
Point Set ◽  
Colored Version ◽  
Special Case

Reay's relaxed Tverberg conjecture and Conway's thrackle conjecture are open problems about the geometry of pairwise intersections. Reay asked for the minimum number of points in Euclidean $d$-space that guarantees any such point set admits a partition into $r$ parts, any $k$ of whose convex hulls intersect. Here we give new and improved lower bounds for this number, which Reay conjectured to be independent of $k$. We prove a colored version of Reay's conjecture for $k$ sufficiently large, but nevertheless $k$ independent of dimension $d$. Pairwise intersecting convex hulls have severely restricted combinatorics. This is a higher-dimensional analogue of Conway's thrackle conjecture or its linear special case. We thus study convex-geometric and higher-dimensional analogues of the thrackle conjecture alongside Reay's problem and conjecture (and prove in two special cases) that the number of convex sets in the plane is bounded by the total number of vertices they involve whenever there exists a transversal set for their pairwise intersections. We thus isolate a geometric property that leads to bounds as in the thrackle conjecture. We also establish tight bounds for the number of facets of higher-dimensional analogues of linear thrackles and conjecture their continuous generalizations.

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