Hamiltonian Cycles Passing Through Prescribed Edges in Locally Twisted Cubes

Let [Formula: see text] be a set of edges whose induced subgraph consists of vertex-disjoint paths in an [Formula: see text]-dimensional locally twisted cube [Formula: see text]. In this paper, we prove that if [Formula: see text] contains at most [Formula: see text] edges, then [Formula: see text] contains a Hamiltonian cycle passing through every edge of [Formula: see text], where [Formula: see text]. [Formula: see text] has a Hamiltonian cycle passing through at most one prescribed edge.

Performance Evaluation, Comparison and Identification of Efficient Hypercube Interconnection Networks

A Multiprocessor is a system with at least two processing units sharing access to memory. The principle goal of utilizing a multiprocessor is to process the undertakings all the while and support the system’s performance. An Interconnection Network interfaces the various handling units and enormously impacts the exhibition of the whole framework. Interconnection Networks, also known as Multi-stage Interconnection Networks, are node-to-node links in which each node may be a single processor or a group of processors. These links transfer information from one processor to the next or from the processor to the memory, allowing the task to be isolated and measured equally. Hypercube systems are a kind of system geography used to interconnect various processors with memory modules and precisely course the information. Hypercube systems comprise of 2n nodes. Any Hypercube can be thought of as a graph with nodes and edges, where a node represents a processing unit and an edge represents a connection between the processors to transmit. Degree, Speed, Node coverage, Connectivity, Diameter, Reliability, Packet loss, Network cost, and so on are some of the different system scales that can be used to measure the performance of Interconnection Networks. A portion of the variations of Hypercube Interconnection Networks include Hypercube Network, Folded Hypercube Network, Multiple Reduced Hypercube Network, Multiply Twisted Cube, Recursive Circulant, Exchanged Crossed Cube Network, Half Hypercube Network, and so forth. This work assesses the performing capability of different variations of Hypercube Interconnection Networks. A group of properties is recognized and a weight metric is structured utilizing the distinguished properties to assess the performance exhibition. Utilizing this weight metric, the performance of considered variations of Hypercube Interconnection Networks is evaluated and summed up to recognize the effective variant. A compact survey of a portion of the variations of Hypercube systems, geographies, execution measurements, and assessment of the presentation are examined in this paper. Degree and Diameter are considered to ascertain the Network cost. On the off chance that Network Cost is considered as the measurement to assess the exhibition, Multiple Reduced Hypercube stands ideal with its lower cost. Notwithstanding it, on the off chance that we think about some other properties/ scales/metrics to assess the performance, any variant other than MRH may show considerably more ideal execution. The considered properties probably won't be ideally adequate to assess the effective performance of Hypercube variations in all respects. On the off chance that a sensibly decent number of properties are utilized to assess the presentation, a proficient variation of Hypercube Interconnection Network can be distinguished for a wide scope of uses. This is the inspiration to do this research work.

Minimum Linear Arrangement of the Cartesian Product of Optimal Order Graph and Path

The minimum linear arrangement of an arbitrary graph is the embedding of the vertices of the graph onto the line topology in such a way that the sum of the distances between adjacent vertices in the graph is minimized. This minimization can be attained by finding an optimal ordering of the vertex set of the graph and labeling the vertices of the line in that order. In this paper, we compute the minimum linear arrangement of the Cartesian product of certain sequentially optimal order graphs which include interconnection networks such as hypercube, folded hypercube, complete Josephus cube and locally twisted cube with path and the edge faulty counterpart of sequentially optimal order graphs.

Grossberg–Karshon Twisted Cubes and Hesitant Jumping Walk Avoidance

Let $G$ be a complex simply-laced semisimple algebraic group of rank $r$ and $B$ a Borel subgroup. Let $\mathbf i \in [r]^n$ be a word and let $\boldsymbol{\ell} = (\ell_1,\dots,\ell_n)$ be a sequence of non-negative integers. Grossberg and Karshon introduced a virtual lattice polytope associated to $\mathbf i$ and $\boldsymbol{\ell}$ called a twisted cube, whose lattice points encode the character of a $B$-representation. More precisely, lattice points in the twisted cube, counted with sign according to a certain density function, yields the character of the generalized Demazure module determined by $\mathbf i$ and $\boldsymbol{\ell}$. In recent work, the author and Harada described precisely when the Grossberg–Karshon twisted cube is untwisted, i.e., the twisted cube is a closed convex polytope, in the situation when the integer sequence $\boldsymbol{\ell}$ comes from a weight $\lambda$ of $G$. However, not every integer sequence $\boldsymbol{\ell}$ comes from a weight of $G$. In the present paper, we interpret the untwistedness of Grossberg–Karshon twisted cubes associated with any word $\mathbf i$ and any integer sequence $\boldsymbol{\ell}$ using the combinatorics of $\mathbf i$ and $\boldsymbol{\ell}$. Indeed, we prove that the Grossberg–Karshon twisted cube is untwisted precisely when $\mathbf i$ is hesitant-jumping-$\boldsymbol{\ell}$-walk-avoiding.

Structural Refinement and Density Functional Theory Study of Synthetic Ge-Akaganéite (β-FeOOH)

In this study, we propose a revised structural model for highly ordered synthetic Ge-akaganéite, a stable analogue of tunnel-type Fe-oxyhydroxide, based on the Rietveld refinement of synchrotron X-ray diffraction data and density functional theory with dispersion correction (DFT-D) calculations. In the proposed crystal structure of Ge-akaganéite, Ge is found not only in the tunnel sites as GeO(OH)3− tetrahedra, but also 4/5 of total Ge atoms are in the octahedral sites substituting 1/10 of Fe. In addition, the tunnel structures are stabilized by the presence of hydrogen bonds between the framework OH and Cl− species, forming a twisted cube structure and the GeO(OH)3− tetrahedra corner oxygen, forming a conjugation bond. The chemical formula of the synthetic Ge-akaganéite was determined to be (Fe7.2Ge0.8)O8.8(OH)7.2Cl0.8(Ge(OH)4)0.2.

Communication Performance Evaluation of the Locally Twisted Cube

The topology properties of multi-processors interconnection networks are important to the performance of high performance computers. The hypercube network [Formula: see text] has been proved to be one of the most popular interconnection networks. The [Formula: see text]-dimensional locally twisted cube [Formula: see text] is an important variant of [Formula: see text]. Fault diameter and wide diameter are two communication performance evaluation parameters of a network. Let [Formula: see text]), [Formula: see text] and [Formula: see text] denote the diameter, the [Formula: see text] fault diameter and the wide diameter of [Formula: see text], respectively. In this paper, we prove that [Formula: see text] if [Formula: see text] is an odd integer with [Formula: see text], [Formula: see text] if [Formula: see text] is an even integer with [Formula: see text].

Three Types of Two-Disjoint-Cycle-Cover Pancyclicity and Their Applications to Cycle Embedding in Locally Twisted Cubes

A graph $G=(V,E)$ is two-disjoint-cycle-cover $[r_1,r_2]$-pancyclic if for any integer $l$ satisfying $r_1 \leq l \leq r_2$, there exist two vertex-disjoint cycles $C_1$ and $C_2$ in $G$ such that the lengths of $C_1$ and $C_2$ are $l$ and $|V(G)| - l$, respectively, where $|V(G)|$ denotes the total number of vertices in $G$. On the basis of this definition, we further propose Ore-type conditions for graphs to be two-disjoint-cycle-cover vertex/edge $[r_1,r_2]$-pancyclic. In addition, we study cycle embedding in the $n$-dimensional locally twisted cube $LTQ_n$ under the consideration of two-disjoint-cycle-cover vertex/edge pancyclicity.

A note on minimum linear arrangement for BC graphs

A linear arrangement is a labeling or a numbering or a linear ordering of the vertices of a graph. In this paper, we solve the minimum linear arrangement problem for bijective connection graphs (for short BC graphs) which include hypercubes, Möbius cubes, crossed cubes, twisted cubes, locally twisted cube, spined cube, [Formula: see text]-cubes, etc., as the subfamilies.