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In this paper, a new sub-family of Hypercubes called the Lucas-run graphs R n l are introduced. The name of this new family of graphs is identified with the interesting fact that | V (R n l) | is equal to the n-th Lucas number. The definition of Lucas-run graph is motivated by the Fibonacci-run graph (Eǧecioǧlu and Iršič in Discrete Appl Math 295:70–84, 2021a; in Discrete Appl Math 300:56–71, 2021b). Various interesting structural and enumerative properties of Lucas-run graphs are investigated, including the analogue of the fundamental recursion, number of vertices and edges, radius, center, diameter and medianicity. Some future research directions and open problems concerning Lucas-run graphs are also proposed. [ABSTRACT FROM AUTHOR]

Chemical analyses of powdered rocks by different laboratories often yield varying results, requiring estimation of the rock's true composition and associated uncertainty. Challenges arise from the peculiar nature of geochemical data. Traditionally, major and trace elements have been measured using different methods, resulting in chemical analyses where the sum of the parts fluctuates around 1 rather than precisely totaling 1. Additionally, all chemical analyses contain an undisclosed mass fraction representing undetected chemical elements. Because of this undisclosed and unknown mass fraction, geochemical data represent a particular kind of compositional data in which closure to unity is not guaranteed. We argue that chemical analyses exist in the hypercube while being sampled from a true composition residing in the simplex. Therefore, we propose an algorithm that generates random chemical analyses by simulating the data acquisition protocol in geochemistry. Using the algorithm's output, we measure the bias and mean squared error (MSE) of various estimators of the true mean composition. Additionally, we explore the impact of missing values on estimator performance. Our findings reveal that the optimized binary log-ratio mean, a new estimator, exhibits the lowest MSE and bias. It performs well even with up to 70% missing values, in contrast to other classical estimators such as the arithmetic mean or the geometric mean. Applying our approach to the GeoPT database, which contains replicate analyses of igneous rocks from numerous geochemical laboratories, we introduce an outlier detection technique based on the Mahalanobis distance between a laboratory's logit coordinates and the optimized mean estimate. This enables a probabilistic ranking of laboratories based on the atypicality of their performance. Finally, we offer an accessible R implementation of our findings through the GitHub repository linked to this paper [subject classification numbers: 10 (compositions) 85 (statistics)]. [ABSTRACT FROM AUTHOR]

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]

Segmentations are partitions of an ordered set into non-overlapping intervals. The Consensus Segmentation or Segmentation Aggregation problem is a special case of the median problems with applications in time series analysis and computational biology. A wide range of dissimilarity measures for segmentations can be expressed in terms of potentials, a special type of set-functions. In this contribution, we shed more light on the properties of potentials, and how such properties affect the solutions of the Consensus Segmentation problem. In particular, we disprove a conjecture stated in 2021, and we provide further insights into the theoretical foundations of the problem. [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]

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]

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]

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]

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]

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 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]

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]

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]

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]

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]

The connectivity of a network is an important indicator to its reliability and fault tolerability. Since the faulty elements in the network may have some special structures, two new kinds of conditional connectivity, called h-restricted H-structure connectivity and h-restricted H-substructure connectivity, are proposed as a generalization of conditional connectivity, where h ≥ 1 , and H is some special structure. In this paper, we establish both h-restricted H-structure connectivity and h-restricted H-substructure connectivity for the hypercube Q n , where the special structures are K 1 , K 1 , 1 , K 1 , 2 , respectively. [ABSTRACT FROM AUTHOR]

The anti-Kekulé number of a connected graph G is the smallest number of edges whose deletion results in a connected subgraph having no Kekulé structures (perfect matchings). As a common generalization of (conditional) matching preclusion number and anti-Kekulé number of a graph G, we introduce s-restricted matching preclusion number of G as the smallest number of edges whose deletion results in a subgraph without perfect matchings such that each component has at least s + 1 vertices. In this paper, we first show that conditional matching preclusion problem and anti-Kekulé problem are NP-complete, respectively, then generalize this result to s-restricted matching preclusion problem. Moreover, we give some sufficient conditions to compute s-restricted matching preclusion numbers of regular graphs. As applications, s-restricted matching preclusion numbers of complete graphs, hypercubes and hyper Petersen networks are determined. [ABSTRACT FROM AUTHOR]

Petri nets have been used extensively in behavior modeling of finite state machines for a very long time since it is more expressible and gives parallelism in the designing of models. The problem of analysis in concurrent systems has been a field of interest for many researchers. Still, only reachability and its systematic enumeration were proven to be useful because of simplified concepts and easy applicability. This article shows the reachability in concurrent nets and topological properties reflected in its reachability analysis. It also discusses how a reachability tree is reduced to a graph by homomorphism and how we can find multi-dimensional topological networks with a class of Petri nets. [ABSTRACT FROM AUTHOR]

Given a graph G, the general position problem is to find a largest set S of vertices of G such that no three vertices of S lie on a common geodesic. Such a set is called a gp -set of G, and its cardinality is the gp -number, gp (G) , of G. In this paper, the edge general position problem is introduced as the edge analogue of the general position problem.The edge general position number, gp e (G) , is the size of a largest edge general position set of G. For r-dimensional hypercube Q r , it is proved that gp e (Q r) = 2 r , and for arbitrary tree T, it is shown that gp e (T) is the number of its leaves. The value of gp e (P r □ P s) is determined for every r , s ≥ 2 . To derive these results, the theory of partial cubes is used. Mulder's meta-conjecture on median graphs is also discussed along the way. [ABSTRACT FROM AUTHOR]

We consider the problem of extending and avoiding partial edge colorings of hypercubes; that is, given a partial edge coloring φ of the d-dimensional hypercube Q d , we are interested in whether there is a proper d-edge coloring of Q d that agrees with the coloring φ on every edge that is colored under φ ; or, similarly, if there is a proper d-edge coloring that disagrees with φ on every edge that is colored under φ . In particular, we prove that for any d ≥ 1 , if φ is a partial d-edge coloring of Q d , then φ is avoidable if every color appears on at most d/8 edges and the coloring satisfies a relatively mild structural condition, or φ is proper and every color appears on at most d - 2 edges. We also show that φ is avoidable if d is divisible by 3 and every color class of φ is an induced matching. Moreover, for all 1 ≤ k ≤ d , we characterize for which configurations consisting of a partial coloring φ of d - k edges and a partial coloring ψ of k edges, there is an extension of φ that avoids ψ . [ABSTRACT FROM AUTHOR]

This paper considers a method for detecting abnormal behavior in cyber-physical systems, Internet of Things (IoT) systems, and distributed technological process automated control system ACS by predicting and analyzing of multidimensional time series by means of neuroevolutionary algorithms based on the development of the hypercube substrate. The method is based on the detection of deviations between the current values of the state of the cyber-physical system and the predicted results. The results of studies of the described method demonstrate the correctness and accuracy of its work. [ABSTRACT FROM AUTHOR]

We propose a new puzzle: Label the eight vertices of a cube using distinct integers between 0 and 12 (both inclusive) such that the induced labeling of each edge, given by the sum of the labels of its end points, causes the 12 edges to be labeled with distinct odd numbers 1, 3, ... 23. Any solution to the puzzle is called a super odd-sum labeling. We deftly discover all the super odd-sum labeling of the cube. [ABSTRACT FROM AUTHOR]

We establish some results on the Banach–Mazur distance in small dimensions. Specifically, we determine the Banach–Mazur distance between the cube and its dual (the cross-polytope) in R3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {R}^3$$\end{document} and R4\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {R}^4$$\end{document}. In dimension three this distance is equal to 95\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\frac{9}{5}$$\end{document}, and in dimension four, it is equal to 2. These findings confirm well-known conjectures, which were based on numerical data. Additionally, in dimension two, we use the asymmetry constant to provide a geometric construction of a family of convex bodies that are equidistant to all symmetric convex bodies. [ABSTRACT FROM AUTHOR]

A space bounded O (d / log d) -competitive hypercube packing algorithm with one active bin only is presented. As a starting point we give a simple 1-space bounded hypercube packing algorithm with competitive ratio (3 / 2) d + O ((21 / 16) d) , for d ≥ 3. [ABSTRACT FROM AUTHOR]

Network cost is equal to degree × diameter and is one of the important measurements when evaluating graphs. Torus and hypercube are very well-known graphs. When these graphs expand, a Torus has an advantage in that its degree does not increase. A hypercube has a shorter diameter than that of other graphs, because when the graph expands, the diameter increases by 1. Hypercube Qn has 2n nodes, and its diameter is n. We propose the rotational binary graph (RBG), which has the advantages of both hypercube and Torus. RBGn has 2n nodes and a degree of 4. The diameter of RBGn would be 1.5n + 1. In this paper, we first examine the topology properties of RBG. Second, we construct a binary spanning tree in RBG. Third, we compare other graphs to RBG considering network cost specifically. Fourth, we suggest a broadcast algorithm with a time complexity of 2n − 2. Finally, we prove that RBGn embedded into hypercube Qn results in dilation n, and expansion 1, and congestion 7. [ABSTRACT FROM AUTHOR]

In this paper we disprove a result on the PBIB designs arising from γ bc -sets in Q 4 , with m = 4 association schemes proved by Chaluvaraju et al. (Afr Math 29:47–63, 2018). In Ref. [11], the authors had posed an open problem to find the PBIB designs in hypercubes of dimension higher than 5. Here, we settle the problem for the first few values, and give a general iterative proof technique for higher dimensional hypercubes. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with multivariate Gegenbauer approximation of functions defined in the d-dimensional hypercube. Two new and sharper bounds for the coefficients of multivariate Gegenbauer expansion of analytic functions are presented based on two different extensions of the Bernstein ellipse. We then establish an explicit error bound for the multivariate Gegenbauer approximation associated with an ℓq ball index set in the uniform norm. We also consider the multivariate approximation of functions with finite regularity and derive the associated error bound on the full grid in the uniform norm. As an application, we extend our arguments to obtain some new tight bounds for the coefficients of tensorized Legendre expansions in the context of polynomial approximation of parametrized PDEs. [ABSTRACT FROM AUTHOR]

The strong geodetic number, sg (G) , of a graph G is the smallest number of vertices such that by fixing a suitable geodesic between each pair of selected vertices, all vertices of the graph are covered. In this paper, the formula for sg (K n , m) is given, as well as a formula for the crown graphs S n 0 . Bounds on the strong geodetic number of the hypercube Q n are also discussed. [ABSTRACT FROM AUTHOR]

The surface of a 3-dimensional cube can be continuously flattened onto any of its faces, by moving creases to change the shapes of some faces successively, following Sabitov's volume preserving theorem. Let C n be an n-dimensional cube with n ≥ 4 , and S be the set of its 2-dimensional faces, i.e., the 2-dimensional skeleton of the square faces in C n . We show that S can be continuously flattened onto any face F of S, such that the faces of S that are parallel to F, do not have any crease, that is, they are rigid during the motion. [ABSTRACT FROM AUTHOR]

We consider the problem of existence of a Hamiltonian cycle containing a matching and avoiding some edges in an n-cube Qn; and obtain the following results. Let n ⩾ 3; M ⊂ E(Qn); and F ⊂ E(Qn)M with 1 ⩽ |F| ⩽ 2n − 4 − |M|: If M is a matching and every vertex is incident with at least two edges in the graph Qn − F; then all edges of M lie on a Hamiltonian cycle in Qn − F: Moreover, if |M| = 1 or |M| = 2; then the upper bound of number of faulty edges tolerated is sharp. Our results generalize the well-known result for |M| = 1 [ABSTRACT FROM AUTHOR]

We present a simple construction of an acute set of size 2 d - 1 + 1 in R d for any dimension d. That is, we explicitly give 2 d - 1 + 1 points in the d-dimensional Euclidean space with the property that any three points form an acute triangle. It is known that the maximal number of such points is less than 2 d . Our result significantly improves upon a recent construction, due to Dmitriy Zakharov, with size of order φ d where φ = (1 + 5) / 2 ≈ 1.618 is the golden ratio. [ABSTRACT FROM AUTHOR]

We consider the sum of the coordinates of a simple random walk on the K-dimensional hypercube and prove a double asymptotic of this process, as both the time parameter n and the space parameter K tend to infinity. Depending on the asymptotic ratio of the two parameters, the rescaled processes converge toward either a "stationary Brownian motion," an Ornstein–Uhlenbeck process or a Gaussian white noise. [ABSTRACT FROM AUTHOR]

Given a graph G, the exact distance-p graph G [ ♮ p ] has V(G) as its vertex set, and two vertices are adjacent whenever the distance between them in G equals p. We present formulas describing the structure of exact distance-p graphs of the Cartesian, the strong, and the lexicographic product. We prove such formulas for the exact distance-2 graphs of direct products of graphs. We also consider infinite grids and some other product structures. We characterize the products of graphs of which exact distance graphs are connected. The exact distance-p graphs of hypercubes Q n are also studied. As these graphs contain generalized Johnson graphs as induced subgraphs, we use some known constructions of their colorings. These constructions are applied for colorings of the exact distance-p graphs of hypercubes with the focus on the chromatic number of Q n [ ♮ p ] for p ∈ { n - 2 , n - 3 , n - 4 } . [ABSTRACT FROM AUTHOR]

This paper investigates properties of fragmentary structures and establishes a relation between them and marked acyclic digraphs with one source and also a correspondence between classes of isomorphic fragmentary structures and unmarked acyclic digraphs of certain type, which are called feasible graphs. The concepts of a dimension of a feasible graph and its corresponding isomorphic fragmentary structures are defined. An expression is obtained for the lower-bound estimate of a dimension. A theorem on properties of feasible graphs is proved. The numbers of fragmentary structures and classes of isomorphic fragmentary structures of small dimensions are counted. [ABSTRACT FROM AUTHOR]

In this paper, we consider accessibility percolation on hypercubes, i.e., we place i.i.d. uniform [0, 1] random variables on vertices of a hypercube, and study whether there is a path connecting two vertices such that the values of these random variables increase along the path. We establish a sharp phase transition depending on the difference of the values at the two endpoints and determine the critical window of the phase transition. Our result completely resolves a conjecture of Berestycki et al. (Accessibility percolation with backsteps. Preprint, available at http://arxiv.org/abs/1401.6894). [ABSTRACT FROM AUTHOR]

In the 1-in port model, every vertex of a synchronous network can receive at most one message in each time unit. We consider simultaneous broadcasting of multiple messages from the same source or from distinct sources in such networks with an additional restriction that every received message can be sent out to neighbors only in the next time unit and never to already informed vertex. We use a general concept of level-disjoint partitions developed for this scenario. Here we introduce a subgraph extension technique for efficient spreading information within this concept. Surprisingly, this approach with so called biwheels leads to simultaneous broadcasting of optimal number of messages on a wide class of graphs in optimal time. In particular, we provide tight results for bipartite tori, meshes, hypercubes, Knödel graphs, circulant graphs. We also propose several open problems and conjectures. [ABSTRACT FROM AUTHOR]

The partition graph of a positive integer n, Pn, is the graph whose vertices are the cyclic compositions of n and two vertices are adjacent if one composition is obtained from the other one by replacing two cyclically consecutive parts by their sum. In this paper we introduce and investigate the notions of singular cyclic composition and singular edge of Pn. We associate with every singular edge and every cycle of Pn, whose vertices are aperiodic cyclic compositions of n, a cycle or a set of disjoint cycles of equal length of the hypercube Qn. [ABSTRACT FROM AUTHOR]

Let n≥d≥ℓ≥1 be integers, and denote the n-dimensional hypercube by Qn. A coloring of the ℓ-dimensional subcubes Qℓ in Qn is called a Qℓ-coloring. Such a coloring is d-polychromatic if every Qd in the Qn contains a Qℓ of every color. In this paper we consider a specific class of Qℓ-colorings that are called linear. Given ℓ and d, let plinℓ(d) be the largest number of colors such that there is a d-polychromatic linear Qℓ-coloring of Qn for all n≥d. We prove that for all d≥3, plind-1(d)=2. In addition, using a computer search, we determine plinℓ(d) for some specific values of ℓ and d, in some cases improving on previously known lower bounds. [ABSTRACT FROM AUTHOR]

This study proposes an interactively recurrent self-evolving fuzzy cerebellar model articulation controller (IRSFCMAC) classifier to solve face detection problems. The learning methods of the proposed classifier are based on simultaneous structure and parameter learning. The structure learning is used to decide the proper input space partition, while the parameter learning is based on gradient descent method. The online structure learning does not need to set any initial structure in advance. In other words, the online structure learning algorithm enables the network along of the problem to efficiently identify the required network structure. The advantages of our proposed IRSFCMAC classifier include (1) using a non-constant differentiable Gaussian basis function to model the hypercube structure; (2) applying an interactively recurrent structure to serve as external loops and internal feedbacks by feeding the hypercube cell (rule) firing strength to itself and other hypercube cells (rules); and (3) requiring fewer computing memory. Finally, experimental results show that the proposed IRSFCMAC classifier is a more adaptive and effective face detection than the other classifiers. [ABSTRACT FROM AUTHOR]

A dominating set D of a connected graph G=(V,E) is said to be bi-connected dominating set if the induced subgraphs of both ⟨D⟩ and ⟨V-D⟩ are connected. The bi-connected domination number γbc(G) is the minimum cardinality of a bi-connected dominating set. A γbc-set is a minimum bi-connected dominating set of G. In this paper, we obtain the Partially Balanced Incomplete Block (PBIB)-designs with m = 1, 2, 3, 4 and ⌊p2⌋ association schemes arising from γbc-sets of some special classes of graphs. [ABSTRACT FROM AUTHOR]

Using the approximation theory notions of polynomial mesh and Dubiner distance in a compact set, we derive error estimates for total degree polynomial optimization on Chebyshev grids of the hypercube. [ABSTRACT FROM AUTHOR]

In this paper, decomposition methods for multidimensional data hypercubes of OLAP systems are investigated. Criteria for reducing the computational complexity of the decomposition methods are presented and comparisons are made with the traditional solutions of multidimensional data analysis problems. Examples illustrating the application of these criteria to investigating the dynamics of computational complexity changes for specific types of reduction problems are considered. [ABSTRACT FROM AUTHOR]

0-1 multilinear programming (MP) captures the essence of pattern generation in logical analysis of data (LAD). This paper utilizes graph theoretic analysis of data to discover useful neighborhood properties among data for data reduction and multi-term linearization of the common constraint of an MP pattern generation model in a small number of stronger valid inequalities. This means that, with a systematic way to more efficiently generating Boolean logical patterns, LAD can be used for more effective analysis of data in practice. Mathematical properties and the utility of the new valid inequalities are illustrated on small examples and demonstrated through extensive experiments on 12 real-life data mining datasets. [ABSTRACT FROM AUTHOR]

In this paper we consider some special characteristics of distances between vertices in the $$n$$ -dimensional hypercube graph $$Q_n$$ and, as a consequence, the corresponding symmetry properties of its resolving sets. It is illustrated how these properties can be implemented within a simple greedy heuristic in order to find efficiently an upper bound of the so called metric dimension $$\beta (Q_n)$$ of $$Q_n$$ , i.e. the minimal cardinality of a resolving set in $$Q_n$$ . This heuristic was applied to generate upper bounds of $$\beta (Q_n)$$ for $$n$$ up to $$22$$ , which are for $$n\ge 19$$ better than the existing ones. Starting from these new bounds, some existing upper bounds for $$23\le n\le 90$$ are improved by a dynamic programming procedure. [ABSTRACT FROM AUTHOR]

The n-cube graph is the graph on the vertex set of n-tuples of 0s and 1s, with two vertices joined by an edge if and only if the n-tuples differ in exactly one component. We compute the Smith group of this graph, or, equivalently, the elementary divisors of an adjacency matrix of the graph. [ABSTRACT FROM AUTHOR]

We consider Glauber dynamics for the low-temperature, ferromagnetic Ising Model on the n-dimensional hypercube. We derive precise asymptotic results for the crossover time (the time it takes for the dynamics to go from the configuration with a ' $$-1$$ ' at every vertex, to the configuration with a ' $$+1$$ ' at each vertex) in the limit as the inverse temperature $$\beta \rightarrow \infty $$ . [ABSTRACT FROM AUTHOR]

Gluing of two pseudo functors has been studied by Deligne, Ayoub, and others in the construction of extraordinary direct image functors in étale cohomology, stable homotopy, and mixed motives of schemes. In this article, we study more generally the gluing of finitely many pseudo functors. Given pseudo functors $$F_i:\mathcal {A}_i\rightarrow \mathcal {D}$$ defined on sub-2-categories $$\mathcal {A}_i$$ of a 2-category $$\mathcal {C}$$ , we are concerned with the problem of finding pseudo functors $$\mathcal {C}\rightarrow \mathcal {D}$$ extending $$F_i$$ up to pseudo natural equivalences. With the help of n-fold categories, we organize gluing data for n pseudo functors into 2-categories. We establish general criteria for equivalence between such 2-categories for n pseudo functors and for $$n-1$$ pseudo functors, which can be applied inductively to the gluing problem. Results of this article are used in Zheng (Sci China Math 58(3):565-632, 2015) to construct extraordinary direct image functors in étale cohomology of Deligne-Mumford stacks. [ABSTRACT FROM AUTHOR]

A (0, 2)-graph is a connected graph, where each pair of vertices has either 0 or 2 common neighbours. These graphs constitute a subclass of (0, λ)-graphs introduced by Mulder in 1979. A rectagraph, well known in diagram geometry, is a triangle-free (0, 2)-graph. (0, 2)-graphs include hypercubes, folded cube graphs and some particular graphs such as icosahedral graph, Shrikhande graph, Klein graph, Gewirtz graph, etc. In this paper, we give some local properties of 4-cycles in (0, λ)-graphs and more specifically in (0, 2)-graphs, leading to new characterizations of rectagraphs and hypercubes. [ABSTRACT FROM AUTHOR]

We propose a new peer-to-peer voice-over-IP (VoIP) protocol, called Darkcube, which builds upon a k-hypercube topology presenting interesting properties regarding multi-party VoIP conferencing. This new topology is based on multiple hypercubes isolating the speaker nodes into a main high speed hypercube and efficiently retransmitting to the remaining listener nodes guaranteeing a good conversational voice communication quality among all users and achieving a great overall user experience. Standard hypercube topologies can present certain performance problems when the number of participants is high; our proposal isolates speakers within a main, low delay, hypercube retransmitting pre-mixed audio to the remaining listening hypercubes improving overall performance and supporting more nodes with better quality than Skype under similar conditions. It is tolerant to common problems on current wireless networks such as high network latencies or temporary network outages, as well as malfunctioning or malicious nodes, making it well suited for its operation on mobile devices. [ABSTRACT FROM AUTHOR]

The emergency vehicle location problem to determine the number of ambulance vehicles and their locations satisfying a required reliability level is investigated in this study. This is a complex nonlinear issue involving critical decision making that has inherent stochastic characteristics. This paper studies an iterative optimization algorithm with parameter estimation to solve the emergency vehicle location problem. In the suggested algorithm, a linear model determines the locations of ambulances, while a hypercube simulation is used to estimate and provide parameters regarding ambulance locations. First, we suggest an iterative hypercube optimization algorithm in which interaction parameters and rules for the hypercube and optimization are identified. The interaction rules employed in this study enable our algorithm to always find the locations of ambulances satisfying the reliability requirement. We also propose an iterative simulation optimization algorithm in which the hypercube method is replaced by a simulation, to achieve computational efficiency. The computational experiments show that the iterative simulation optimization algorithm performs equivalently to the iterative hypercube optimization. The suggested algorithms are found to outperform existing algorithms suggested in the literature. [ABSTRACT FROM AUTHOR]