scholarly journals ON STRUCTURAL AND GRAPH THEORETIC PROPERTIES OF HIGHER ORDER DELAUNAY GRAPHS

2009 ◽  
Vol 19 (06) ◽  
pp. 595-615 ◽  
Author(s):  
MANUEL ABELLANAS ◽  
PROSENJIT BOSE ◽  
JESÚS GARCÍA ◽  
FERRAN HURTADO ◽  
CARLOS M. NICOLÁS ◽  
...  
Keyword(s):  
Cellular Networks ◽  
Lower Bounds ◽  
Higher Order ◽  
Combinatorial Structure ◽  
Upper And Lower Bounds ◽  
Coloring Problem ◽  
Graph Theoretic ◽  
Convex Position ◽  
Gabriel Graph ◽  
Vertex Set

Given a set P of n points in the plane, the order-k Delaunay graph is a graph with vertex set P and an edge exists between two points p, q ∈ P when there is a circle through p and q with at most k other points of P in its interior. We provide upper and lower bounds on the number of edges in an order-k Delaunay graph. We study the combinatorial structure of the set of triangulations that can be constructed with edges of this graph. Furthermore, we show that the order-k Delaunay graph is connected under the flip operation when k ≤ 1 but not necessarily connected for other values of k. If P is in convex position then the order-k Delaunay graph is connected for all k ≥ 0. We show that the order-k Gabriel graph, a subgraph of the order-k Delaunay graph, is Hamiltonian for k ≥ 15. Finally, the order-k Delaunay graph can be used to efficiently solve a coloring problem with applications to frequency assignments in cellular networks.

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 LS-category of moment-angle manifolds and higher order Massey products

2021 ◽  
Vol 33 (5) ◽  
pp. 1179-1205
Author(s):  
Piotr Beben ◽  
Jelena Grbić
Keyword(s):  
Simplicial Complex ◽  
Structural Properties ◽  
Lower Bounds ◽  
Higher Order ◽  
Hyperbolic Manifolds ◽  
Upper And Lower Bounds ◽  
Connected Sum ◽  
Massey Products ◽  
Higher Dimensional ◽  
Lusternik Schnirelmann

Abstract Using the combinatorics of the underlying simplicial complex K, we give various upper and lower bounds for the Lusternik–Schnirelmann (LS) category of moment-angle complexes 𝒵 K {\mathcal{Z}_{K}} . We describe families of simplicial complexes and combinatorial operations which allow for a systematic description of the LS-category. In particular, we characterize the LS-category of moment-angle complexes 𝒵 K {\mathcal{Z}_{K}} over triangulated d-manifolds K for d ≤ 2 {d\leq 2} , as well as higher-dimensional spheres built up via connected sum, join, and vertex doubling operations. We show that the LS-category closely relates to vanishing of Massey products in H * ⁢ ( 𝒵 K ) {H^{*}(\mathcal{Z}_{K})} , and through this connection we describe first structural properties of Massey products in moment-angle manifolds. Some of the further applications include calculations of the LS-category and the description of conditions for vanishing of Massey products for moment-angle manifolds over fullerenes, Pogorelov polytopes and k-neighborly complexes, which double as important examples of hyperbolic manifolds.

scholarly journals Online Graph Coloring Against a Randomized Adversary

2018 ◽  
Vol 29 (04) ◽  
pp. 551-569 ◽  
Author(s):  
Elisabet Burjons ◽  
Juraj Hromkovič ◽  
Rastislav Královič ◽  
Richard Královič ◽  
Xavier Muñoz ◽  
...  
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Online Algorithms ◽  
Graph Coloring ◽  
Competitive Ratio ◽  
Upper And Lower Bounds ◽  
Coloring Problem ◽  
Single Instance ◽  
Graph Coloring Problem

We consider an online model where an adversary constructs a set of [Formula: see text] instances [Formula: see text] instead of one single instance. The algorithm knows [Formula: see text] and the adversary will choose one instance from [Formula: see text] at random to present to the algorithm. We further focus on adversaries that construct sets of [Formula: see text]-chromatic instances. In this setting, we provide upper and lower bounds on the competitive ratio for the online graph coloring problem as a function of the parameters in this model. Both bounds are linear in [Formula: see text] and matching upper and lower bound are given for a specific set of algorithms that we call “minimalistic online algorithms”.

scholarly journals Antipodal graphs and digraphs

1993 ◽  
Vol 16 (3) ◽  
pp. 579-586 ◽  
Author(s):  
Garry Johns ◽  
Karen Sleno
Keyword(s):  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Vertex Set

The antipodal graph of a graphG, denoted byA(G), has the same vertex set asGwith an edge joining verticesuandvifd(u,v)is equal to the diameter ofG. (IfGis disconnected, thendiam G=∞.) This definition is extended to a digraphDwhere the arc(u,v)is included inA(D)ifd(u,v)is the diameter ofD. It is shown that a digraphDis an antipodal digraph if and only ifDis the antipodal digraph of its complement. This generalizes a known characterization for antipodal graphs and provides an improved proof. Examples and properties of antipodal digraphs are given. A digraphDis self-antipodal ifA(D)is isomorphic toD. Several characteristics of a self-antipodal digraphDare given including sharp upper and lower bounds on the size ofD. Similar results are given for self-antipodal graphs.

scholarly journals Fully-Dynamic and Kinetic Conflict-Free Coloring of Intervals with Respect to Points

2019 ◽  
Vol 29 (01) ◽  
pp. 49-72
Author(s):  
Mark de Berg ◽  
Tim Leijsen ◽  
Aleksandar Markovic ◽  
André van Renssen ◽  
Marcel Roeloffzen ◽  
...  
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Worst Case ◽  
Insertions And Deletions ◽  
Coloring Problem ◽  
Trade Offs ◽  
Special Cases ◽  
The Cost ◽  
Case Number

We introduce the fully-dynamic conflict-free coloring problem for a set [Formula: see text] of intervals in [Formula: see text] with respect to points, where the goal is to maintain a conflict-free coloring for [Formula: see text] under insertions and deletions. A coloring is conflict-free if for each point [Formula: see text] contained in some interval, [Formula: see text] is contained in an interval whose color is not shared with any other interval containing [Formula: see text]. We investigate trade-offs between the number of colors used and the number of intervals that are recolored upon insertion or deletion of an interval. Our results include: a lower bound on the number of recolorings as a function of the number of colors, which implies that with [Formula: see text] recolorings per update the worst-case number of colors is [Formula: see text], and that any strategy using [Formula: see text] colors needs [Formula: see text] recolorings; a coloring strategy that uses [Formula: see text] colors at the cost of [Formula: see text] recolorings, and another strategy that uses [Formula: see text] colors at the cost of [Formula: see text] recolorings; stronger upper and lower bounds for special cases. We also consider the kinetic setting where the intervals move continuously (but there are no insertions or deletions); here we show how to maintain a coloring with only four colors at the cost of three recolorings per event and show this is tight.

scholarly journals Analysis of directional higher order jump discontinuities with trigonometric shearlets

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Kevin Schober ◽  
Jürgen Prestin
Keyword(s):  
Lower Bounds ◽  
Recent Article ◽  
Higher Order ◽  
Upper And Lower Bounds ◽  
Characteristic Functions ◽  
Directional Derivatives ◽  
Periodic Functions ◽  
Decay Properties ◽  
Jump Discontinuities ◽  
Periodic Characteristic

<p style='text-indent:20px;'>In a recent article, we showed that trigonometric shearlets are able to detect directional step discontinuities along edges of periodic characteristic functions. In this paper, we extend these results to bivariate periodic functions which have jump discontinuities in higher order directional derivatives along edges. In order to prove suitable upper and lower bounds for the shearlet coefficients, we need to generalize the results about localization- and orientation-dependent decay properties of the corresponding inner products of trigonometric shearlets and the underlying periodic functions.</p>

On the Complexity of Computing and Learning with Multiplicative Neural Networks

2002 ◽  
Vol 14 (2) ◽  
pp. 241-301 ◽  
Author(s):  
Michael Schmitt
Keyword(s):  
Neural Networks ◽  
Lower Bounds ◽  
Higher Order ◽  
Upper And Lower Bounds ◽  
Biological Behavior ◽  
Vc Dimension ◽  
Neuron Models ◽  
Learning Capabilities ◽  
Order Of Magnitude ◽  
The Impact

In a great variety of neuron models, neural inputs are combined using the summing operation. We introduce the concept of multiplicative neural networks that contain units that multiply their inputs instead of summing them and thus allow inputs to interact nonlinearly. The class of multiplicative neural networks comprises such widely known and well-studied network types as higher-order networks and product unit networks. We investigate the complexity of computing and learning for multiplicative neural networks. In particular, we derive upper and lower bounds on the Vapnik-Chervonenkis (VC) dimension and the pseudo-dimension for various types of networks with multiplicative units. As the most general case, we consider feedforward networks consisting of product and sigmoidal units, showing that their pseudo-dimension is bounded from above by a polynomial with the same order of magnitude as the currently best-known bound for purely sigmoidal networks. Moreover, we show that this bound holds even when the unit type, product or sigmoidal, may be learned. Crucial for these results are calculations of solution set components bounds for new network classes. As to lower bounds, we construct product unit networks of fixed depth with super-linear VC dimension. For sigmoidal networks of higher order, we establish polynomial bounds that, in contrast to previous results, do not involve any restriction of the network order. We further consider various classes of higher-order units, also known as sigma-pi units, that are characterized by connectivity constraints. In terms of these, we derive some asymptotically tight bounds. Multiplication plays an important role in both neural modeling of biological behavior and computing and learning with artificial neural networks. We briefly survey research in biology and in applications where multiplication is considered an essential computational element. The results we present here provide new tools for assessing the impact of multiplication on the computational power and the learning capabilities of neural networks.

Sign in / Sign up

Export Citation Format

Share Document