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.

scholarly journals Design and Analysis of a Symmetric Log Star Graph with a Smaller Network Cost Than Star Graphs

Electronics ◽  
2021 ◽  
Vol 10 (8) ◽  
pp. 981
Author(s):  
Jung-Hyun Seo ◽  
Hyeong-Ok Lee
Keyword(s):  
Parallel Computing ◽  
Computer Science ◽  
Regular Graph ◽  
Connected Graph ◽  
Routing Algorithm ◽  
Star Graph ◽  
Topological Properties ◽  
Star Graphs ◽  
Network Cost ◽  
Network Costs

Graphs are used as models to solve problems in fields such as mathematics, computer science, physics, and chemistry. In particular, torus, hypercube, and star graphs are popular when modeling the connection structure of processors in parallel computing because they are symmetric and have a low network cost. Whereas a hypercube has a substantially smaller diameter than a torus, star graphs have been presented as an alternative to hypercubes because of their lower network cost. We propose a novel log star (LS) that is symmetric and has a lower network cost than a star graph. The LS is an undirected, recursive, and regular graph. In LSn, the number of nodes is n! while the degree is 2log2n − 1 and the diameter is 0.5n(log2n)2 + 0.75nlog2n. In this study, we analyze the basic topological properties of LS. We prove that LSn is a symmetrical connected graph and analyzed its subgraph characteristics. Then, we propose a routing algorithm and derive the diameter and network cost. Finally, the network costs of the LS and star graph-like networks are compared.

scholarly journals Cycle partitions of regular graphs

2020 ◽  
pp. 1-24
Author(s):  
Vytautas Gruslys ◽  
Shoham Letzter
Keyword(s):  
Regular Graph ◽  
Bipartite Graphs ◽  
Disjoint Paths ◽  
Regular Graphs ◽  
Simple Construction ◽  
Vertex Set ◽  
Almost All ◽  
Vertex Disjoint

Abstract Magnant and Martin conjectured that the vertex set of any d-regular graph G on n vertices can be partitioned into $n / (d+1)$ paths (there exists a simple construction showing that this bound would be best possible). We prove this conjecture when $d = \Omega(n)$ , improving a result of Han, who showed that in this range almost all vertices of G can be covered by $n / (d+1) + 1$ vertex-disjoint paths. In fact our proof gives a partition of V(G) into cycles. We also show that, if $d = \Omega(n)$ and G is bipartite, then V(G) can be partitioned into n/(2d) paths (this bound is tight for bipartite graphs).

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.

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.

scholarly journals Optimal Construction of Edge-Disjoint Paths in Random Regular Graphs

2000 ◽  
Vol 9 (3) ◽  
pp. 241-263 ◽  
Author(s):  
ALAN M. FRIEZE ◽  
LEI ZHAO
Keyword(s):  
Polynomial Time ◽  
Regular Graph ◽  
Randomized Algorithm ◽  
Constant Factor ◽  
Disjoint Paths ◽  
Regular Graphs ◽  
Edge Disjoint ◽  
Random Regular Graph ◽  
Arbitrary Graphs ◽  
Optimal Construction

Given a graph G = (V, E) and a set of κ pairs of vertices in V, we are interested in finding, for each pair (ai, bi), a path connecting ai to bi such that the set of κ paths so found is edge-disjoint. (For arbitrary graphs the problem is [Nscr ][Pscr ]-complete, although it is in [Pscr ] if κ is fixed.)We present a polynomial time randomized algorithm for finding edge-disjoint paths in the random regular graph Gn,r, for sufficiently large r. (The graph is chosen first, then an adversary chooses the pairs of end-points.) We show that almost every Gn,r is such that all sets of κ = Ω(n/log n) pairs of vertices can be joined. This is within a constant factor of the optimum.

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.

Node-to-set disjoint paths problem in star graphs

1997 ◽  
Vol 62 (4) ◽  
pp. 201-207 ◽  
Author(s):  
Qian-Ping Gu ◽  
Shietung Peng
Keyword(s):  
Disjoint Paths ◽  
Star Graphs

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>

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.

Sign in / Sign up

Export Citation Format

Share Document