scholarly journals On the Nodal Structure of Nonlinear Stationary Waves on Star Graphs

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 185
Author(s):  
Ram Band ◽  
Sven Gnutzmann ◽  
August Krueger
Keyword(s):  
Existence Of Solutions ◽  
Stationary Waves ◽  
Star Graph ◽  
Oscillation Theorem ◽  
Nodal Domains ◽  
Nodal Structure ◽  
Star Graphs ◽  
Spectral Curves ◽  
Nonlinear Quantum ◽  
Cubic Nonlinear Schrödinger Equation

We consider stationary waves on nonlinear quantum star graphs, i.e., solutions to the stationary (cubic) nonlinear Schrödinger equation on a metric star graph with Kirchhoff matching conditions at the centre. We prove the existence of solutions that vanish at the centre of the star and classify them according to the nodal structure on each edge (i.e., the number of nodal domains or nodal points that the solution has on each edge). We discuss the relevance of these solutions in more applied settings as starting points for numerical calculations of spectral curves and put our results into the wider context of nodal counting, such as the classic Sturm oscillation theorem.

Diagnosability of Bubble-Sort Star Graphs with Missing Edges

2019 ◽  
Vol 19 (02) ◽  
pp. 1950002 ◽  
Author(s):  
SHIYING WANG ◽  
YINGYING WANG
Keyword(s):  
Multiprocessor System ◽  
Star Graph ◽  
Star Graphs ◽  
Comparison Model ◽  
Model Following ◽  
Strong Property

The diagnosability of a multiprocessor system plays an important role. The bubble-sort star graph BSn has many good properties. In this paper, we study the diagnosis on BSn under the comparison model. Following the concept of the local diagnosability, the strong local diagnosability property is discussed. This property describes the equivalence of the local diagnosability of a node and its degree. We prove that BSn (n ≥ 5) has this property, and it keeps this strong property even if there exist (2n − 5) missing edges in it, and the result is optimal with respect to the number of missing edges.

EMBEDDING OF CYCLES AND GRIDS IN STAR GRAPHS

1991 ◽  
Vol 01 (01) ◽  
pp. 43-74 ◽  
Author(s):  
JUNG-SING JWO ◽  
S. LAKSHMIVARAHAN ◽  
S. K. DHALL
Keyword(s):  
Hamiltonian Cycle ◽  
Parallel Computers ◽  
Star Graph ◽  
Attractive Feature ◽  
Star Graphs ◽  
New Class ◽  
Number Of Authors

The use of the star graph as a viable interconnection scheme for parallel computers has been examined by a number of authors in recent times. An attractive feature of this class of graphs is that it has sublogarithmic diameter and has a great deal of symmetry akin to the binary hypercube. In this paper we describe a new class of algorithms for embedding (a) Hamiltonian cycle (b) the set of all even cycles and (c) a variety of two- and multi-dimensional grids in a star graph. In addition, we also derive an algorithm for the ranking and the unranking problem with respect to the Hamiltonian cycle.

scholarly journals Forcing operators on star graphs applied for the cubic fourth order Schrödinger equation

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Roberto de A. Capistrano–Filho ◽  
Márcio Cavalcante ◽  
Fernando A. Gallego
Keyword(s):  
Schrödinger Equation ◽  
Fourth Order ◽  
Schrodinger Equation ◽  
Initial Boundary ◽  
Common Vertex ◽  
Star Graph ◽  
Star Graphs ◽  
Low Regularity ◽  
Fourier Restriction ◽  
Starting Point

<p style='text-indent:20px;'>In a recent article [<xref ref-type="bibr" rid="b16">16</xref>], the authors gave a starting point of the study on a series of problems concerning the initial boundary value problem and control theory of Biharmonic NLS in some non-standard domains. In this direction, this article deals to present answers for some questions left in [<xref ref-type="bibr" rid="b16">16</xref>] concerning the study of the cubic fourth order Schrödinger equation in a star graph structure <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{G} $\end{document}</tex-math></inline-formula>. Precisely, consider <inline-formula><tex-math id="M2">\begin{document}$ \mathcal{G} $\end{document}</tex-math></inline-formula> composed by <inline-formula><tex-math id="M3">\begin{document}$ N $\end{document}</tex-math></inline-formula> edges parameterized by half-lines <inline-formula><tex-math id="M4">\begin{document}$ (0,+\infty) $\end{document}</tex-math></inline-formula> attached with a common vertex <inline-formula><tex-math id="M5">\begin{document}$ \nu $\end{document}</tex-math></inline-formula>. With this structure the manuscript proposes to study the well-posedness of a dispersive model on star graphs with three appropriated vertex conditions by using the <i>boundary forcing operator approach</i>. More precisely, we give positive answer for the Cauchy problem in low regularity Sobolev spaces. We have noted that this approach seems very efficient, since this allows to use the tools of Harmonic Analysis, for instance, the Fourier restriction method, introduced by Bourgain, while for the other known standard methods to solve partial differential partial equations on star graphs are more complicated to capture the dispersive smoothing effect in low regularity. The arguments presented in this work have prospects to be applied for other nonlinear dispersive equations in the context of star graphs with unbounded edges.</p>

scholarly journals Existence of solutions for a class of nonlinear boundary value problems on the hexasilinane graph

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ali Turab ◽  
Zoran D. Mitrović ◽  
Ana Savić
Keyword(s):  
Boundary Value Problems ◽  
Nonlinear Boundary ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Chemical Formula ◽  
Nonlinear Boundary Value Problems ◽  
Star Graphs ◽  
Chemical Graph ◽  
Chemical Graph Theory ◽  
Fractional Boundary Value Problems

AbstractChemical graph theory is a field of mathematics that studies ramifications of chemical network interactions. Using the concept of star graphs, several investigators have looked into the solutions to certain boundary value problems. Their choice to utilize star graphs was based on including a common point connected to other nodes. Our aim is to expand the range of the method by incorporating the graph of hexasilinane compound, which has a chemical formula $\mathrm{H}_{12} \mathrm{Si}_{6}$ H 12 Si 6 . In this paper, we examine the existence of solutions to fractional boundary value problems on such graphs, where the fractional derivative is in the Caputo sense. Finally, we include an example to support our significant findings.

THE PERMUTATIONAL GRAPH: A NEW NETWORK TOPOLOGY

1998 ◽  
Vol 09 (01) ◽  
pp. 3-11
Author(s):  
SATOSHI OKAWA
Keyword(s):  
Network Topology ◽  
Equivalence Classes ◽  
Star Graph ◽  
Complete Graphs ◽  
Graph Topology ◽  
Star Graphs ◽  
Network Topologies ◽  
Graph Families ◽  
The Relationship ◽  
Better Than

This paper introduces the penmutational graph, a new network topology, which preserves the same desirable properties as those of a star graph topology. A permutational graph can be decomposed into subgraphs induced by node sets defined by equivalence classes. Using this decomposition, its structual properties as well as the relationship among graph families, permutational graphs, star graphs, and complete graphs are studied. Moreover, the diameters of permutational graphs are investigated and good estimates are obtained which are better than those of some network topologies of similar orders.

Optimal Neighborhood Broadcast in Star Graphs

2003 ◽  
Vol 04 (04) ◽  
pp. 419-428 ◽  
Author(s):  
Satoshi Fujita
Keyword(s):  
Interconnection Networks ◽  
Completion Time ◽  
Single Port ◽  
Multicast Tree ◽  
Communication Model ◽  
Star Graph ◽  
Star Graphs ◽  
Multicast Trees ◽  
Multicast Scheme ◽  
Special Case

In this paper, we consider the problem of constructing a multicast tree in star interconnection networks under the single-port communication model. Unlike previous schemes for constructing space-efficient multicast trees, we adopt the completion time of each multicast as the objective function to be minimized. In particular, we study a special case of the problem in which all destination vertices are immediate neighbors of the source vertex, and propose a multicast scheme of [Formula: see text] time units for the star graph of dimension n.

scholarly journals Optimal Bounds for Disjoint Hamilton Cycles in Star Graphs

2018 ◽  
Vol 29 (03) ◽  
pp. 377-389 ◽  
Author(s):  
Parisa Derakhshan ◽  
Walter Hussak
Keyword(s):  
Interconnection Network ◽  
Multiple Edge ◽  
Star Graph ◽  
Hamilton Cycles ◽  
Star Graphs ◽  
Edge Disjoint ◽  
Spanning Subgraph ◽  
Optimal Bounds ◽  
Hamilton Decompositions ◽  
Hamilton Decomposition

In interconnection network topologies, the [Formula: see text]-dimensional star graph [Formula: see text] has [Formula: see text] vertices corresponding to permutations [Formula: see text] of [Formula: see text] symbols [Formula: see text] and edges which exchange the positions of the first symbol [Formula: see text] with any one of the other symbols. The star graph compares favorably with the familiar [Formula: see text]-cube on degree, diameter and a number of other parameters. A desirable property which has not been fully evaluated in star graphs is the presence of multiple edge-disjoint Hamilton cycles which are important for fault-tolerance. The only known method for producing multiple edge-disjoint Hamilton cycles in [Formula: see text] has been to label the edges in a certain way and then take images of a known base 2-labelled Hamilton cycle under different automorphisms that map labels consistently. However, optimal bounds for producing edge-disjoint Hamilton cycles in this way, and whether Hamilton decompositions can be produced, are not known for any [Formula: see text] other than for the case of [Formula: see text] which does provide a Hamilton decomposition. In this paper we show that, for all n, not more than [Formula: see text], where [Formula: see text] is Euler’s totient function, edge-disjoint Hamilton cycles can be produced by such automorphisms. Thus, for non-prime [Formula: see text], a Hamilton decomposition cannot be produced. We show that the [Formula: see text] upper bound can be achieved for all even [Formula: see text]. In particular, if [Formula: see text] is a power of 2, [Formula: see text] has a Hamilton decomposable spanning subgraph comprising more than half of the edges of [Formula: see text]. Our results produce a better than twofold improvement on the known bounds for any kind of edge-disjoint Hamilton cycles in [Formula: see text]-dimensional star graphs for general [Formula: see text].

Artinian Local Rings Whose Annihilating-ideal Graphs Are Star Graphs

2015 ◽  
Vol 22 (01) ◽  
pp. 73-82 ◽  
Author(s):  
Houyi Yu ◽  
Tongsuo Wu ◽  
Weiping Gu
Keyword(s):  
Local Ring ◽  
Complete Characterization ◽  
Local Rings ◽  
Star Graph ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Star Graphs ◽  
Finite Local Ring ◽  
Necessary And Sufficient

In this paper, a necessary and sufficient condition is given for a commutative Artinian local ring whose annihilating-ideal graph is a star graph. Also, a complete characterization is established for a finite local ring whose annihilating-ideal graph is a star graph.

THE SPANNING CONNECTIVITY OF THE (n,k)-STAR GRAPHS

2006 ◽  
Vol 17 (02) ◽  
pp. 415-434 ◽  
Author(s):  
HONG-CHUN HSU ◽  
CHENG-KUAN LIN ◽  
HUA-MIN HUNG ◽  
LIH-HSING HSU
Keyword(s):  
Regular Graph ◽  
Disjoint Paths ◽  
Star Graph ◽  
Star Graphs

A k-containerC(u, v) of a graph G is a set of k-disjoint paths joining u to v. A k-container C(u, v) is a k*-container if every vertex of G is incident with a path in C(u, v). A graph G is k*-connected if there exists a k*-container between any two distinct vertices u and v. A k-regular graph G is super spanning connected if G is i*-connected for all 1 ≤ i ≤ k. In this paper, we prove that the (n, k)-star graph Sn,k is super spanning connected if n ≥ 3 and (n-k) ≥ 2.

Parameter ranges for the existence of solutions whose state components have specified nodal structure in coupled multiparameter systems of nonlinear Sturm–Liouville boundary value problems

1991 ◽  
Vol 119 (3-4) ◽  
pp. 347-365 ◽  
Author(s):  
Robert Stephen Cantrell
Keyword(s):  
Boundary Value Problems ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
A Priori ◽  
Preceding Paper ◽  
Analytic Structure ◽  
Nodal Structure ◽  
The Maximum Principle ◽  
Sturm Liouville ◽  
Parameter Ranges

SynopsisThe set of solutions to the two-parameter systemhas been shown in a preceding paper of the author to exhibit a topological-functional analytic structure analogous to the structure of solution sets for nonlinear Sturm–Liouville boundary value problems. As the parameter λ and µ are varied, transitions in the solution set occur, first from trivial solutions to solutions (u, 0) with u having n nodes on (a, b) or solutions (0, v) with v having m nodes on (a, b), and then to solutions of the form (u, v), where u has n nodes on (a, b) and v has m nodes on (a, b), with n possibly different from m. Moreover, each transition is global in an appropriate bifurcation theoretic sense, with preservation of nodal structure. This paper explores these phenomena more closely, focusing on the range of parameters (λ, µ) for the existence of solutions (u, v) with u having n nodes on (a, b) and v having m nodes on (a, b) and its dependence on the assumptions placed on the coupling functions f and g. The principal tools of the analysis are the Alexander–Antman Bifurcation Theorem and a priori estimate techniques based on the maximum principle.

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