Search

The Steiner tree problem in graphs is widely studied because of its usefulness in network design and circuit layout. In this context, given a set of vertices S (| S | ≥ 2 ,) a tree that connects all vertices in S is called an S-Steiner tree. This helps to measure how well a network G can connect any set of S vertices together. In an S-Steiner tree, if each vertex in S has only one connection, it is called a pendant S-Steiner tree. Two pendant S-Steiner trees, T and T ′ , are internally disjoint if E (T) ∩ E (T ′) = ∅ and V (T) ∩ V (T ′) = S. The local pendant tree-connectivity, denoted as τ G (S) , represents the maximum number of internally disjoint pendant S-Steiner trees in graph G. For an integer k with 2 ≤ k ≤ n , where n is the number of vertices, the pendant k-tree-connectivity, denoted as τ k (G) , is defined as τ k (G) = m i n { τ G (S) : S ⊆ V (G) , | S | = k }. This paper focuses on studying the pendant 3-tree-connectivity of augmented cubes, which are modified versions of hypercubes designed to enhance connectivity and reduce diameter. This research demonstrates that the pendant 3-tree-connectivity of augmented cubes, denoted as τ 3 (A Q n) is 2 n - 3 . This result matches the upper bound of τ 3 (G) provided by Hager, specifically for the augmented cube graph A Q n . [ABSTRACT FROM AUTHOR]

Chung et al. constructed an induced subgraph of the hypercube Qn ${Q}^{n}$ with α(Qn)+1 $\alpha ({Q}^{n})+1$ vertices and with maximum degree smaller than ⌈n⌉ $\lceil \sqrt{n}\rceil $. Subsequently, Huang proved the Sensitivity Conjecture by demonstrating that the maximum degree of such an induced subgraph of hypercube Qn ${Q}^{n}$ is at least ⌈n⌉ $\lceil \sqrt{n}\rceil $, and posed the question: Given a graph G $G$, let f(G) $f(G)$ be the minimum of the maximum degree of an induced subgraph of G $G$ on α(G)+1 $\alpha (G)+1$ vertices, what can we say about f(G) $f(G)$? In this paper, we investigate this question for Cartesian product of paths Pm ${P}_{m}$, denoted by Pmk ${P}_{m}^{k}$. We determine the exact values of f(Pmk) $f({P}_{m}^{k})$ when m=2n+1 $m=2n+1$ by showing that f(P2n+1k)=1 $f({P}_{2n+1}^{k})=1$ for n≥2 $n\ge 2$ and f(P3k)=2 $f({P}_{3}^{k})=2$, and give a nontrivial lower bound of f(Pmk) $f({P}_{m}^{k})$ when m=2n $m=2n$ by showing that f(P2nk)≥⌈2cosπn2n+1k⌉ $f({P}_{2n}^{k})\ge \lceil 2\cos \frac{\pi n}{2n+1}\sqrt{k}\rceil $. In particular, when n=1 $n=1$, we have f(Qk)=f(P2k)≥k $f({Q}^{k})=f({P}_{2}^{k})\ge \sqrt{k}$, which is Huang's result. The lower bounds of f(P3k) $f({P}_{3}^{k})$ and f(P2nk) $f({P}_{2n}^{k})$ are given by using the spectral method provided by Huang. [ABSTRACT FROM AUTHOR]

In the year 2000, Eric Egge introduced the generalized Terwilliger algebra T of a distance-regular graph Γ . For any vertex x of Γ , there is a surjective algebra homomorphism ♮ from T to the Terwilliger algebra T(x). If Γ is a complete graph, then ♮ is an isomorphism. If Γ is not complete, then ♮ may or may not be an isomorphism, and in general the details are unknown. We show that if Γ is a hypercube, there exists an isomorphism from T to a direct sum of full matrix algebras. Using this result, we then show that if Γ is a hypercube, the algebra homomorphism ♮ : T → T (x) is an isomorphism for all vertices x of Γ . [ABSTRACT FROM AUTHOR]

The stability, robustness, accuracy, and efficiency of space-time finite element methods crucially depend on the choice of approximation spaces for test and trial functions. This is especially true for high-order, mixed finite element methods which often must satisfy an inf-sup condition in order to ensure stability. With this in mind, the primary objective of this paper and a companion paper is to provide a wide range of explicitly stated, conforming, finite element spaces in four dimensions. In this paper, we construct explicit high-order conforming finite elements on 4-cubes (tesseracts); our construction uses tools from the recently developed 'Finite Element Exterior Calculus'. With a focus on practical implementation, we provide details including Piola-type transformations, and explicit expressions for the volumetric, facet, face, edge, and vertex degrees of freedom. In addition, we establish important theoretical properties, such as the exactness of the finite element sequences, and the unisolvence of the degrees of freedom. [ABSTRACT FROM AUTHOR]

We say that a Hamilton cycle C = (x 1 , ... , x n) in a graph G is k-symmetric, if the mapping x i ↦ x i + n / k for all i = 1 , ... , n , where indices are considered modulo n, is an automorphism of G. In other words, if we lay out the vertices x 1 , ... , x n equidistantly on a circle and draw the edges of G as straight lines, then the drawing of G has k-fold rotational symmetry, i.e., all information about the graph is compressed into a 360 ∘ / k wedge of the drawing. The maximum k for which there exists a k-symmetric Hamilton cycle in G is referred to as the Hamilton compression of G. We investigate the Hamilton compression of four different families of vertex-transitive graphs, namely hypercubes, Johnson graphs, permutahedra and Cayley graphs of abelian groups. In several cases, we determine their Hamilton compression exactly, and in other cases, we provide close lower and upper bounds. The constructed cycles have a much higher compression than several classical Gray codes known from the literature. Our constructions also yield Gray codes for bitstrings, combinations and permutations that have few tracks and/or that are balanced. [ABSTRACT FROM AUTHOR]

For a graph $H$ and a hypercube $Q_n$ , $\textrm{ex}(Q_n, H)$ is the largest number of edges in an $H$ -free subgraph of $Q_n$. If $\lim _{n \rightarrow \infty } \textrm{ex}(Q_n, H)/|E(Q_n)| \gt 0$ , $H$ is said to have a positive Turán density in a hypercube or simply a positive Turán density; otherwise, it has zero Turán density. Determining $\textrm{ex}(Q_n, H)$ and even identifying whether $H$ has a positive or zero Turán density remains a widely open question for general $H$. By relating extremal numbers in a hypercube and certain corresponding hypergraphs, Conlon found a large class of graphs, ones having so-called partite representation, that have zero Turán density. He asked whether this gives a characterisation, that is, whether a graph has zero Turán density if and only if it has partite representation. Here, we show that, as suspected by Conlon, this is not the case. We give an example of a class of graphs which have no partite representation, but on the other hand, have zero Turán density. In addition, we show that any graph whose every block has partite representation has zero Turán density in a hypercube. [ABSTRACT FROM AUTHOR]

Let Qd ${Q}_{d}$ be the hypercube of dimension d $d$ and let H $H$ and K $K$ be subsets of the vertex set V(Qd) $V({Q}_{d})$, called configurations in Qd ${Q}_{d}$. We say that K $K$ is an exact copy of H $H$ if there is an automorphism of Qd ${Q}_{d}$ which sends H $H$ onto K $K$. Let n≥d $n\ge d$ be an integer, let H $H$ be a configuration in Qd ${Q}_{d}$ and let S $S$ be a configuration in Qn ${Q}_{n}$. We let λ(H,d,n) $\lambda (H,d,n)$ be the maximum, over all configurations S $S$ in Qn ${Q}_{n}$, of the fraction of sub‐d $d$‐cubes R $R$ of Qn ${Q}_{n}$ in which S∩R $S\cap R$ is an exact copy of H $H$, and we define the d $d$‐cube density λ(H,d) $\lambda (H,d)$ of H $H$ to be the limit as n $n$ goes to infinity of λ(H,d,n) $\lambda (H,d,n)$. We determine λ(H,d) $\lambda (H,d)$ for several configurations in Q3 ${Q}_{3}$ and Q4 ${Q}_{4}$ as well as for an infinite family of configurations. There are strong connections with the inducibility of graphs. [ABSTRACT FROM AUTHOR]

Population-based meta-heuristics use particles represented by d - dimensional points to encode candidate solutions to an optimization problem. The goal of this work is to introduce a new type of particle represented by a d - dimensional hypercube. Our new representation offers a field of view to swarm particles, making them aware of their surroundings. This gives each particle its own coverage of the search space. We evaluated the effect of our approach in the performance of seven swarm-based algorithms, including artificial bee colony, cuckoo search optimization and particle swarm optimization. We used 30 well-known benchmark functions and six different numbers of dimensions. We compared the performances of the algorithms with blind and sighted particles using Mann-Whitney tests. The results of our extensive experiments show that sighted particles perform at least as well as blind ones in 93% of all scenarios, across different algorithms, benchmark functions and numbers of dimensions, showing significantly better results 51% of the time. We also proposed a new scale which measures an algorithm's exploration and exploitation capabilities and we used it to explain our results. Finally, we followed a sighted swarm throughout the convergence process, showing its ability to cover large portions of the search space. Our take-home message is that any swarm algorithm can adopt our particle representation in order to improve its performance, because any method can benefit from improving its exploitation and exploration capabilities. [ABSTRACT FROM AUTHOR]

For a simple connected graph G, the generalized distance matrix Dα(G) is defined as Dα(G) = αTr(G) + (1 - α)D(G), 0 ≤ α ≤ 1. The largest eigenvalue of Dα(G) is called the generalized distance spectral radius or Dα-spectral radius of G. In this paper, we obtain some upper bounds for the generalized distance spectral radius in terms of various graph parameters associated with the structure of graph G, and characterize the extremal graphs attaining these bounds. We determine the graphs with minimal generalized distance spectral radius among the trees with given diameter d and among all unicyclic graphs with given girth. We also obtain the generalized distance spectrum of the square of the cycle and the square of the hypercube of dimension n. We show that the square of the hypercube of dimension n has three distinct generalized distance eigenvalues. [ABSTRACT FROM AUTHOR]

Designing a cost-effective, energy-efficient and highly scalable network for data centers that can deliver sufficient bandwidth has drawn tremendous attentions recently. The data center networks constructed by using multi-port servers can provide sufficient bandwidth, such as BCube, DCell and GBC3. As the volume of data keeps growing rapidly, more and more servers are continuously added into data centers. In order to reduce the cost and energy consumption, data center networks can be expanded gradually by adding a small number of servers from time to time instead of adding a huge number of servers at a time. This paper proposes a new type of data center network structure called GHB, which is constructed by using commercial switches and multi-port servers. Two types of incomplete GHB structures are also proposed. A small number of servers can be gradually added into the incomplete structures without changing their topological properties. As shown in the experimental results, the throughput of GHB is comparable to that of BCube, and is larger than that of GBC3 and DCell. The analysis results indicate that GHB strikes a good balance among diameter, bisection width, incremental scalability, cost, and energy consumption in contrast to BCube, DCell and GBC3. Compared with the BCube and GBC3, GHB reduces the cost and energy consumption by about 6% and 18%, respectively. The highest throughput of GHB is higher than that of DCell and GBC3 by about 3.4% and 8.35%, respectively. [ABSTRACT FROM AUTHOR]

Motivated by a multitude of practical applications, many distinct vulnerability parameters of multiprocessor systems have been explored. Traditional connectivity and diagnosability are undoubtedly the most well investigated of these metrics, but often fail to capture the most subtle differences of a multiprocessor system. Subsequently, it is necessary to take into account the minimum degree of components, the size of components or the number of components. However, the structure of the components is ignored in these circumstances. In this work, we propose a novel diagnostic strategy based on cyclic connectivity, namely the cyclic diagnosability. The cyclic diagnosability, denoted by |$ct(G)$|⁠ , is the maximum size of the faulty vertex set |$F$| of |$G$| such that the self-diagnosable system |$G$| can identify all the vertices in |$F$| under the condition that at least two connected components of |$G-F$| contain a cycle. Furthermore, we investigate the cyclic diagnosability of hypercube |$Q_{n}$| under the PMC model and the MM* model, and show that |$ct(Q_{n})=5n-10$| for |$n\geq 7$|⁠. [ABSTRACT FROM AUTHOR]

The undirected graph, exchanged hypercube |$EH(s,t)$|⁠ , is a variant of hypercube proposed by Loh et al. It is obtained by removing some links from |$(s+t+1)$| -dimensional hypercube. It retains many excellent properties, so many people have studied its reliability and fault tolerance. In this paper, combining the structure connectivity and substructure connectivity of graphs proposed not long ago, we obtain its |$P_k$| -path, |$C_{2l}$| -cycle and |$K_{1,r}$| -star structure connectivity and substructure connectivity where |$2\le k,r\le s-1\le t-1$| and |$6\le 2l\le s-1\le t-1$|⁠ ; we also establish |$\kappa ^s(EH(s,t);C_4)$| for |$5\le s\le t$| and the upper bound of |$\kappa (EH(s,t);C_4)$| for |$4\le s\le t$|⁠. [ABSTRACT FROM AUTHOR]

We give a short constructive proof for the existence of a Hamilton cycle in the subgraph of the (2 n + 1) -dimensional hypercube induced by all vertices with exactly n or n + 1 many 1s. [ABSTRACT FROM AUTHOR]

The quad-cube is a special case of the metacube that itself is derivable from the hypercube. It is amenable to an application as a network topology, especially when the node size exceeds several million. This paper presents a sequence of graphs, leading up to the quad-cube, where the graphs in the sequence are important in their own right. For example, they exhibit hypercube-like low diameters and efficient domination parameters, and most of them admit a Hamiltonian cycle. The hierarchy is likely to be useful in the future. [ABSTRACT FROM AUTHOR]

Using the sawtooth map as the basis of a coupled map lattice enables simple analytic results to be obtained for the global Lyapunov spectra of a number of standard lattice networks. The results presented can be used to enrich a course on chaos or dynamical systems by providing tractable examples of higher dimensional maps and links to a number of standard results in matrices. A number of suggestions for classroom use are given. [ABSTRACT FROM AUTHOR]

An element i = (v, e) of a graph G is called an incidence of G, if v ∈ V (G), e ∈ E(G) and v ∈ e. A vi-simultaneous proper coloring of a graph G is a coloring of the vertices and incidences of G properly at the same time such that any two adjacent or incident elements receive distinct colors. In this paper, we investigate the simultaneous coloring of vertices and incidences of hypercubes. [ABSTRACT FROM AUTHOR]

A connected graph G is called l 1 -embeddable, if it can be isometrically embedded into the l 1 -space. The shifted quadrilateral cylinder graph O m , n , k is a class of quadrilateral tilings on a cylinder obtained by rolling the grid graph P m □ P n via shifting k positions. In this article, we determine that all the O m , n , k are not l 1 -embeddable except for O m , n , 0 and O m , 3 , 1 . [ABSTRACT FROM AUTHOR]

In this paper, a brand new heuristic optimization algorithm for multiple-valued logic (MVL) functions and the performance of the implementation will be introduced. In logic synthesis, optimization procedure is a fundamental process in order to design more efficient systems. The proposed algorithm, called MVL-MIN, utilizes Cube Algebra operators. MVL-MIN computes the prime cubes that cover a target cube at a time, instead of extracting all the primes for a given MVL function. MVL-MIN consists of five major procedures, called (a) checking, (b) expansion, (c) elimination, (d) nondisjoint sharp, and (e) prime determination. In order to find a near-minimal cover, three different prime determination functions are developed. The implementation based on the new algorithm is compared with MVSIS. For testing, three sets of test bench are prepared. Each set includes 5 MVL functions. Test results proved that MVL-MIN is able to solve all test files within a fixed allocation of computer resources while MVSIS could not. MVL-MIN achieved 1.9× to 9.7× speedup over MVSIS. In terms of cover size, since current MVL-MIN implementation cannot recognize redundant cubes, MVSIS computed more concise covers for all test benches than MVL-MIN. [ABSTRACT FROM AUTHOR]

Blood vessel visualization technology allows nursing staff to transition from traditional palpation or touch to locate the subcutaneous blood vessels to visualized localization by providing a clear visual aid for performing various medical procedures accurately and efficiently involving blood vessels; this can further improve the first-attempt puncture success rate for nursing staff and reduce the pain of patients. We propose a novel technique for hyperspectral visualization of blood vessels in human skin. An experiment with six participants with different skin types, race, and nationality backgrounds is described. A mere separation of spectral layers for different skin types is shown to be insufficient. The use of three-wavelength indices in imaging has shown a significant improvement in the quality of results compared to using only two-wavelength indices. This improvement can be attributed to an increase in the contrast ratio, which can be as high as 25%. We propose and implement a technique for finding new index formulae based on an exhaustive search and a binary blood-vessel image obtained through an expert assessment. As a result of the search, a novel index formula was deduced, allowing high-contrast blood vessel images to be generated for any skin type. [ABSTRACT FROM AUTHOR]

An exchanged hypercube is a spanning subgraph of a hypercube. It retains a large number of desirable properties of the hypercube, yet maintains a reduced interconnection complexity. This paper shows that the graph is isomorphic to an induced subgraph of the hypercube of the least possible size. Further, the minimal hypercube consists of two factors, each of which comprises as many vertex-disjoint copies of the induced subgraph. The result is seamlessly inherited by the dual-cube that is known to be a special case of the exchanged hypercube. [ABSTRACT FROM AUTHOR]

A set D of vertices in an isolate-free graph is a total dominating set if every vertex is adjacent to a vertex in D. If the set D has the additional property that the subgraph induced by D contains a perfect matching, then D is a paired dominating set of G. The total domination number γt(G) and the paired domination number γpr(G) of a graph G are the minimum cardinalities of a total dominating set and a paired dominating set of G, respectively. The total domination stability (respectively, paired domination stability) of G, denoted stγt(G) (respectively, stγpr(G)), is the minimum size of a non-isolating set of vertices in G whose removal changes the total domination number (respectively, paired domination number). In this paper, we study total and paired domination stability in prisms. [ABSTRACT FROM AUTHOR]

In this paper we study maximal chains in certain lattices constructed from powers of chains by iterated lax colimits in the 2-category of posets. Such a study is motivated by the fact that in lower dimensions, we get some familiar combinatorial objects such as Dyck paths and Kreweras walks. [ABSTRACT FROM AUTHOR]

At present, the reliability of interconnection networks of multiprocessing systems has become a hot topic of research concern for parallel computer systems. Conditional connectivity is an important parameter to measure the reliability of an interconnected network. In reality, the failure of one node will inevitably have a negative impact on the surrounding nodes. Often it is the specific structures that fail in an interconnected network. Therefore, we propose two novel kinds of connectivity, called g-extra H-structure connectivity and g-extra H-substructure connectivity, to go for a more accurate measure of the reliability of the network. Hypercube network is the most dominant interconnection network topology used by computer systems today, for example, the famous parallel computing systems Cray T3D, Cray T3E, IBM Blue Gene, etc. are built with it as the interconnection network topology. In this paper, we obtain the results of the g-extra H-structure connectivity and the g-extra H-substructure connectivity of the hypercubes when the specific structure is Pk and g = 1. [ABSTRACT FROM AUTHOR]

At present, the reliability of interconnection networks of multiprocessing systems has become a hot topic of research concern for parallel computer systems. Conditional connectivity is an important parameter to measure the reliability of an interconnected network. In reality, the failure of one node will inevitably have a negative impact on the surrounding nodes. Often it is the specific structures that fail in an interconnected network. Therefore, we propose two novel kinds of connectivity, called $ g $-extra $ H $-structure connectivity and $ g $-extra $ H $-substructure connectivity, to go for a more accurate measure of the reliability of the network. Hypercube network is the most dominant interconnection network topology used by computer systems today, for example, the famous parallel computing systems Cray $ T3D $, Cray $ T3E $, $ IBM $ Blue Gene, etc. are built with it as the interconnection network topology. In this paper, we obtain the results of the $ g $-extra $ H $-structure connectivity and the $ g $-extra $ H $-substructure connectivity of the hypercubes when the specific structure is $ P_k $ and $ g = 1 $.

Using the special value at u = 1 of the Artin-Ihara L-function, we give a short proof of the count of the number of spanning trees in the n-cube. [ABSTRACT FROM AUTHOR]

Parallel structure and switching circuit have become a fundamental theme in the concern of Artificial Intelligence and interconnection network and also it is revealed to be critical when researching in high performance. This Study helps to the process of breaking down larger process into normal form and independent secondary module, which helps to perform operations with various properties such as atomicity, consistency; isolation and switching features. They also help for manage and tune processing during faults in architecture, the results of which are combined upon completion as part of an overall algorithm. In this paper firstly we select the independent small module of hypercube then represent into data structure and apply logical operations into vectors. [ABSTRACT FROM AUTHOR]

The quad-cube is a special case of the metacube that itself is derivable from the hypercube. It is amenable to an application as a network topology, especially when the node size exceeds several million. This paper presents the following welcome properties of the graph, relating to its structure: (1) vertex transitivity that facilitates the working of an algorithm meant for a "local" context in the global context as well, and (2) an exact formula for the distance metric, which leads to a precise result on the distance-wise vertex distribution of the graph and an exact formula for the average vertex distance. Remarkably, the vertex distribution of the quad-cube resembles, to a large extent, the vertex distribution of the twin copies of a hypercube. In a parallel study, the author recently reported similar results with respect to the dual-cube (Jha in J Supercomput 78:17758–17775, 2022) [ABSTRACT FROM AUTHOR]

One of the important issues in evaluating an interconnection network is to study the hamiltonian cycle embedding problems. A graph G is spanning k-edge-cyclable if for any k independent edges e 1 , e 2 , ... , e k of G, there exist k vertex-disjoint cycles C 1 , C 2 , ... , C k in G such that V (C 1) ∪ V (C 2) ∪ ⋯ ∪ V (C k) = V (G) and e i ∈ E (C i) for all 1 ≤ i ≤ k . According to the definition, the problem of finding hamiltonian cycle focuses on k = 1 . The notion of spanning edge-cyclability can be applied to the problem of identifying faulty links and other related issues in interconnection networks. In this paper, we prove that the n-dimensional hypercube Q n is spanning k-edge-cyclable for 1 ≤ k ≤ n − 1 and n ≥ 2 . This is the best possible result, in the sense that the n-dimensional hypercube Q n is not spanning n-edge-cyclable. [ABSTRACT FROM AUTHOR]

With increasing public demand for ready-to-eat fresh-cut fruit, the postharvest industry requires the development and adaptation of monitoring technologies to provide customers with a product of consistent quality. The fresh-cut trade of pineapples (Ananas comosus) is on the rise, favored by the sensory quality of the product and mechanization of the cutting process. In this paper, a multispectral imaging-based approach is introduced to provide distribution maps of moisture content, soluble solids content, and carotenoids content in fresh-cut pineapple. A dataset containing hyperspectral images (380–1690 nm) and reference measurements in 10 regions of interest of 60 fruit (n = 600) was prepared. Ranking and uncorrelatedness (based on ReliefF algorithm) and subset selection (based on CfsSubset algorithm) approaches were applied to find the most informative wavelengths in which bandpass optical filters or light sources are commercially available. The correlation coefficient and error metrics obtained by cross-validated multilayer perceptron neural network models indicated that the superior selected wavelengths (495, 500, 505, 1215, 1240, and 1425 nm) resulted in prediction of moisture content with R = 0.56, MAPE = 1.92%, soluble solids content with R = 0.52, MAPE = 14.72%, and carotenoids content with R = 0.63, MAPE = 43.99%. Prediction of chemical composition in each pixel of the multispectral images using the calibration models yielded spatially distributed quantification of the fruit slice, spatially varying according to the maturation of single fruitlets in the whole pineapple. Calibration models provided reliable responses spatially throughout the surface of fresh-cut pineapple slices with a constant error. According to the approach to use commercially relevant wavelengths, calibration models could be applied in classifying fruit segments in the mechanized preparation of fresh-cut produce. [ABSTRACT FROM AUTHOR]

Copyright of Operations Research Transactions / Yunchouxue Xuebao is the property of Editorial office of Operations Research Transactions and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Fat‐tree topologies are widely used in interconnect network designs for parallel supercomputers. In the classic fat‐tree, compute nodes are connected to leaf stage switches by links. Given a large number of compute nodes, many switches and links are required, resulting in high hardware costs. To solve this problem, this paper proposes two hybrid topologies, k$$ k $$‐Cube k$$ k $$‐Ary n$$ n $$‐Tree (CAT) and Mirrored k$$ k $$‐Cube k$$ k $$‐Ary n$$ n $$‐Tree (MiCAT), based on fat‐tree and hypercube. Instead of connecting k$$ k $$ compute nodes directly to a leaf switch, we connect a k$$ k $$‐cube to the switch, and each switch in the k$$ k $$‐cube part connects k$$ k $$ compute nodes. That is, this k$$ k $$‐cube consists of 2k−1$$ {2}^k-1 $$ switches and k(2k−1)$$ k\left({2}^k-1\right) $$ compute nodes. We give the shortest path routing algorithms and evaluate the path diversity, cost, performance, and average packet latency of CAT and MiCAT. The results show that CAT and MiCAT can save up to 87%$$ 87\% $$ switches and 80%$$ 80\% $$ links in a large‐scale parallel system, k=n=8$$ k=n=8 $$ for example, compared to fat‐trees, and meanwhile, both CAT and MiCAT have higher path diversities than fat‐trees. [ABSTRACT FROM AUTHOR]

For a given graph G, the general position problem asks for the largest set of vertices S ⊆ V (G) , such that no three distinct vertices of S belong to a common shortest path of G. The general position problem for Cartesian products of two cycles as well as for hypercubes is considered. The problem is completely solved for the first family of graphs, while for the hypercubes, some partial results based on reduction to SAT are given. [ABSTRACT FROM AUTHOR]

The engineering of new states of matter through Floquet driving has revolutionized the field of condensed matter physics. This technique enables the creation of hybrid topological states and ordered phases that are absent in normal systems. Crystalline structures, exemplifying spatially ordered systems under periodic driving, have been extensively studied. However, recent focus has shifted towards discrete time crystals (DTCs), periodically driven quantum many-body systems that break time translation symmetry under specific conditions. In this paper, the model of discrete time crystals is extended to allow for the formation of time-varying tesseracts, allowing for the investigation of time translational symmetry in pseudo-higher-dimensional lattice systems. [ABSTRACT FROM AUTHOR]

The Wiener index and the Mostar index quantify two distance related properties of connected graphs: the Wiener index is the sum of the distances over all pairs of vertices and the Mostar index is a measure of how far the graph is from being distance-balanced. These two measures have been considered for a number of interesting families of graphs. In this paper, we determine the Wiener index and the Mostar index of alternate Lucas cubes. Alternate Lucas cubes form a family of interconnection networks whose recursive construction mimics the construction of the well-known Fibonacci cubes.

The conditional h-vertex (h-edge) connectivity of a connected graph H of minimum degree k > h is the size of a smallest vertex (edge) set F of H such that H − F is a disconnected graph of minimum degree at least h. Let G be the Cartesian product of r ≥ 1 cycles, each of length at least four and let h be an integer such that 0 ≤ h ≤ 2r − 2. In this paper, we determine the conditional h-vertex-connectivity and the conditional h-edge-connectivity of the graph G. We prove that both these connectivities are equal to (2r−h)arh, where arh is the number of vertices of a smallest h-regular subgraph of G.

Let ex (Q n , H) be the largest number of edges in a subgraph G of a hypercube Q n such that there is no subgraph of G isomorphic to H. We show that for any integer k ≥ 3 ex (Q n , C 4 k + 2) = O (n 5 6 + 1 3 (2 k − 2) 2 n). [ABSTRACT FROM AUTHOR]

As the volume of data keeps growing rapidly, more and more storage devices, servers, and network devices are continuously added into data center networks (DCNs) to store, manage, and analyze the data. The industry experience indicates that, instead of adding a huge number of servers into the DCNs at a time, the DCN can also be expanded gradually by adding a small number of servers from time to time. This paper proposes a new type of dual-centric data center network structure, called GHDC (Generalized Hypercube Data Center network architecture), which is constructed by using commodity switches and multi-port servers. After analyzing the shortest distance between any two vertices, a routing algorithm for GHDC is developed. To achieve incremental scalability, two incomplete GHDC structures are proposed. A small number of servers can be added into the incomplete GHDC structures while their topological properties are maintained. The analysis and experiment results show that GHDC significantly outperform the other DCN structures, such as FatTree, BCube, Platonica, FCell, FSquare, and FRectangle, in terms of the incremental scalability and robustness. The average throughput of GHDC is approximately comparable to that of FatTree, BCube, and FSquare, and is higher than that of Platonica, FCell, and FRectangle by about 130.2%, 17.45% and 25.5%. Compared with the FatTree, BCube, FCell, FRectangle, and FSquare, GHDC reduces the cost by about 68.49%, 78.04%, 10.84%, 22.85%, and 29.55%, and reduces the max energy consumption by about 69.45%, 34.48%, 11.55%, 24.31%, and 29.58%, respectively. The actual energy consumption of GHDC is much lower than that of FatTree, BCube, FCell, FRectangle, and FSquare, and is little higher than that of Platonia. [ABSTRACT FROM AUTHOR]

On a descriptive level, this paper presents a number of logical fragments which require the Boolean algebra B 5 , i.e., bitstrings of length five, for their semantic analysis. Two categories from the realm of natural language quantification are considered, namely, proportional quantification with fractions and percentages—as in two thirds/66 percent of the children are asleep—and normative quantification—as in not enough/too many children are asleep. On a more theoretical level, we study two distinct Aristotelian subdiagrams in B 5 , which are the result of moving from B 5 to B 4 either by collapsing bit positions or by deleting bit positions. These two operations are also argued to shed a new light on earlier results from Logical Geometry, in which the collapsing or deletion of bit positions triggers a shift from B 4 to B 3 . [ABSTRACT FROM AUTHOR]