30 research outputs found

    Recursive circulants and their embeddings among hypercubes

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    AbstractWe propose an interconnection structure for multicomputer networks, called recursive circulant. Recursive circulant G(N,d) is defined to be a circulant graph with N nodes and jumps of powers of d. G(N,d) is node symmetric, and has some strong hamiltonian properties. G(N,d) has a recursive structure when N=cdm, 1⩽c<d. We develop a shortest-path routing algorithm in G(cdm,d), and analyze various network metrics of G(cdm,d) such as connectivity, diameter, mean internode distance, and visit ratio. G(2m,4), whose degree is m, compares favorably to the hypercube Qm. G(2m,4) has the maximum possible connectivity, and its diameter is ⌈(3m−1)/4⌉. Recursive circulants have interesting relationship with hypercubes in terms of embedding. We present expansion one embeddings among recursive circulants and hypercubes, and analyze the costs associated with each embedding. The earlier version of this paper appeared in Park and Chwa (Proc. Internat. Symp. Parallel Architectures, Algorithms and Networks ISPAN’94, Kanazawa, Japan, December 1994, pp. 73–80)

    Long cycles and paths in distance graphs

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    AbstractFor n∈N and D⊆N, the distance graph PnD has vertex set {0,1,…,n−1} and edge set {ij∣0≤i,j≤n−1,|j−i|∈D}. Note that the important and very well-studied circulant graphs coincide with the regular distance graphs.A fundamental result concerning circulant graphs is that for these graphs, a simple greatest common divisor condition, their connectivity, and the existence of a Hamiltonian cycle are all equivalent. Our main result suitably extends this equivalence to distance graphs. We prove that for a finite set D of order at least 2, there is a constant cD such that the greatest common divisor of the integers in D is 1 if and only if for every n, PnD has a component of order at least n−cD if and only if for every n≥cD+3, PnD has a cycle of order at least n−cD. Furthermore, we discuss some consequences and variants of this result

    Spherical and Hyperbolic Toric Topology-Based Codes On Graph Embedding for Ising MRF Models: Classical and Quantum Topology Machine Learning

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    The paper introduces the application of information geometry to describe the ground states of Ising models by utilizing parity-check matrices of cyclic and quasi-cyclic codes on toric and spherical topologies. The approach establishes a connection between machine learning and error-correcting coding. This proposed approach has implications for the development of new embedding methods based on trapping sets. Statistical physics and number geometry applied for optimize error-correcting codes, leading to these embedding and sparse factorization methods. The paper establishes a direct connection between DNN architecture and error-correcting coding by demonstrating how state-of-the-art architectures (ChordMixer, Mega, Mega-chunk, CDIL, ...) from the long-range arena can be equivalent to of block and convolutional LDPC codes (Cage-graph, Repeat Accumulate). QC codes correspond to certain types of chemical elements, with the carbon element being represented by the mixed automorphism Shu-Lin-Fossorier QC-LDPC code. The connections between Belief Propagation and the Permanent, Bethe-Permanent, Nishimori Temperature, and Bethe-Hessian Matrix are elaborated upon in detail. The Quantum Approximate Optimization Algorithm (QAOA) used in the Sherrington-Kirkpatrick Ising model can be seen as analogous to the back-propagation loss function landscape in training DNNs. This similarity creates a comparable problem with TS pseudo-codeword, resembling the belief propagation method. Additionally, the layer depth in QAOA correlates to the number of decoding belief propagation iterations in the Wiberg decoding tree. Overall, this work has the potential to advance multiple fields, from Information Theory, DNN architecture design (sparse and structured prior graph topology), efficient hardware design for Quantum and Classical DPU/TPU (graph, quantize and shift register architect.) to Materials Science and beyond.Comment: 71 pages, 42 Figures, 1 Table, 1 Appendix. arXiv admin note: text overlap with arXiv:2109.08184 by other author

    Author index to volumes 41–60 (1981–1984)

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    Maximal diameter of integral circulant graphs

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    Integral circulant graphs are proposed as models for quantum spin networks that permit a quantum phenomenon called perfect state transfer. Specifically, it is important to know how far information can potentially be transferred between nodes of the quantum networks modelled by integral circulant graphs and this task is related to calculating the maximal diameter of a graph. The integral circulant graph ICGn(D)ICG_n (D) has the vertex set Zn={0,1,2,,n1}Z_n = \{0, 1, 2, \ldots, n - 1\} and vertices aa and bb are adjacent if gcd(ab,n)D\gcd(a-b,n)\in D, where D{d:dn, 1d<n}D \subseteq \{d : d \mid n,\ 1\leq d<n\}. Motivated by the result on the upper bound of the diameter of ICGn(D)ICG_n(D) given in [N. Saxena, S. Severini, I. Shparlinski, \textit{Parameters of integral circulant graphs and periodic quantum dynamics}, International Journal of Quantum Information 5 (2007), 417--430], according to which 2D+12|D|+1 represents one such bound, in this paper we prove that the maximal value of the diameter of the integral circulant graph ICGn(D)ICG_n(D) of a given order nn with its prime factorization p1α1pkαkp_1^{\alpha_1}\cdots p_k^{\alpha_k}, is equal to r(n)r(n) or r(n)+1r(n)+1, where r(n)=k+{i αi>1, 1ik}r(n)=k + |\{ i \ | \alpha_i> 1,\ 1\leq i\leq k \}|, depending on whether n∉4N+2n\not\in 4N+2 or not, respectively. Furthermore, we show that, for a given order nn, a divisor set DD with Dk|D|\leq k can always be found such that this bound is attained. Finally, we calculate the maximal diameter in the class of integral circulant graphs of a given order nn and cardinality of the divisor set tkt\leq k and characterize all extremal graphs. We actually show that the maximal diameter can have the values 2t2t, 2t+12t+1, r(n)r(n) and r(n)+1r(n)+1 depending on the values of tt and nn. This way we further improve the upper bound of Saxena, Severini and Shparlinski and we also characterize all graphs whose diameters are equal to 2D+12|D|+1, thus generalizing a result in that paper.Comment: 29 pages, 1 figur

    Hyperdeterminantal computation for the Laughlin wave function

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    International audienceThe decomposition of the Laughlin wave function in the Slater orthogonal basis appears in the discussion on the second-quantized form of the Laughlin states and is straightforwardly equivalent to the decomposition of the even powers of the Vandermonde determinants in the Schur basis. Such a computation is notoriously difficult and the coefficients of the expansion have not yet been interpreted. In our paper, we give an expression of these coefficients in terms of hyperdeterminants of sparse tensors. We use this result to construct an algorithm allowing to compute one coefficient of the development without computing the others. Thanks to a program in {\tt C}, we performed the calculation for the square of the Vandermonde up to an alphabet of eleven lettres

    The directed Oberwolfach problem with variable cycle lengths: a recursive construction

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    The directed Oberwolfach problem OP(m1,,mk)^\ast(m_1,\ldots,m_k) asks whether the complete symmetric digraph KnK_n^\ast, assuming n=m1++mkn=m_1+\ldots +m_k, admits a decomposition into spanning subdigraphs, each a disjoint union of kk directed cycles of lengths m1,,mkm_1,\ldots,m_k. We hereby describe a method for constructing a solution to OP(m1,,mk)^\ast(m_1,\ldots,m_k) given a solution to OP(m1,,m)^\ast(m_1,\ldots,m_\ell), for some <k\ell<k, if certain conditions on m1,,mkm_1,\ldots,m_k are satisfied. This approach enables us to extend a solution for OP(m1,,m)^\ast(m_1,\ldots,m_\ell) into a solution for OP(m1,,m,t)^\ast(m_1,\ldots,m_\ell,t), as well as into a solution for OP(m1,,m,2t)^\ast(m_1,\ldots,m_\ell,2^{\langle t \rangle}), where 2t2^{\langle t \rangle} denotes tt copies of 2, provided tt is sufficiently large. In particular, our recursive construction allows us to effectively address the two-table directed Oberwolfach problem. We show that OP(m1,m2)^\ast(m_1,m_2) has a solution for all 2m1m22 \le m_1\le m_2, with a definite exception of m1=m2=3m_1=m_2=3 and a possible exception in the case that m1{4,6}m_1 \in \{ 4,6 \}, m2m_2 is even, and m1+m214m_1+m_2 \ge 14. It has been shown previously that OP(m1,m2)^\ast(m_1,m_2) has a solution if m1+m2m_1+m_2 is odd, and that OP(m,m)^\ast(m,m) has a solution if and only if m3m \ne 3. In addition to solving many other cases of OP^\ast, we show that when 2m1++mk132 \le m_1+\ldots +m_k \le 13, OP(m1,,mk)^\ast(m_1,\ldots,m_k) has a solution if and only if (m1,,mk)∉{(4),(6),(3,3)}(m_1,\ldots,m_k) \not\in \{ (4),(6),(3,3) \}
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