Symmetric Graphs Research Articles

Error Thresholds for Arbitrary Pauli Noise

The error threshold of a one-parameter family of quantum channels is defined as the largest noise level such that the quantum capacity of the channel remains positive. This in turn guarantees the existence of a quantum error correction code for noise modeled by that channel. Discretizing the single-qubit errors leads to the important family of Pauli quantum channels; curiously, multipartite entangled states can increase the threshold of these channels beyond the so-called hashing bound, an effect termed superadditivity of coherent information. In this work, we divide the simplex of Pauli channels into one-parameter families and compute numerical lower bounds on their error thresholds. We find substantial increases of error thresholds relative to the hashing bound for large regions in the Pauli simplex corresponding to biased noise, which is a realistic noise model in promising quantum computing architectures. The error thresholds are computed on the family of graph states, a special type of stabilizer state. In order to determine the coherent information of a graph state, we devise an algorithm that exploits the symmetries of the underlying graph, resulting in a substantial computational speed-up. This algorithm uses tools from computational group theory and allows us to consider symmetric graph states on a large number of vertices. Our algorithm works particularly well for repetition codes and concatenated repetition codes (or cat codes), for which our results provide the first comprehensive study of superadditivity for arbitrary Pauli channels. In addition, we identify a novel family of quantum codes based on tree graphs. The error thresholds of these tree graph states outperform repetition and cat codes in large regions of the Pauli simplex, and hence form a new code family with desirable error correction properties.

Stability of stationary points for one-dimensional Willmore energy with spatially heterogeneous term

We consider a variational problem that consists of the Willmore energy of planar curves and a linear term in the curvature with a spatially heterogeneous coefficient. This coefficient is assumed to be a piecewise constant function. The variational problem arises from the modeling of the epidermal membrane in human skin. It has been predicted that the spatial heterogeneity of cell adhesion is one of the important factors in the protuberance formation of the membrane. We investigate the existence and stability of stationary points represented by symmetric graphs and find a condition for the existence of such stationary points by using the methods developed in the study of the Navier problem for Willmore energy. Especially, we give a necessary and sufficient condition for the existence of symmetric stationary points in terms of the gap size of the spatial heterogeneity. Moreover, we show some results on the stability of these stationary points.

Generalized Gardiner–Praeger graphs and their symmetries

A subgroup of the automorphism group of a graph acts half-arc-transitively on the graph if it acts transitively on the vertex-set and on the edge-set of the graph but not on the arc-set of the graph. If the full automorphism group of the graph acts half-arc-transitively, the graph is said to be half-arc-transitive.In 1994 Gardiner and Praeger introduced two families of tetravalent arc-transitive graphs, called the C±1 and the C±ε graphs, that play a prominent role in the characterization of the tetravalent graphs admitting an arc-transitive group of automorphisms with a normal elementary abelian subgroup such that the corresponding quotient graph is a cycle. All of the Gardiner–Praeger graphs are arc-transitive but admit a half-arc-transitive group of automorphisms. Quite recently, Potočnik and Wilson introduced the family of CPM graphs, which are generalizations of the Gardiner–Praeger graphs. Most of these graphs are arc-transitive, but some of them are half-arc-transitive. In fact, at least up to order 1000, each tetravalent half-arc-transitive loosely-attached graph of odd radius having vertex-stabilizers of order greater than 2 is isomorphic to a CPM graph.In this paper we determine the automorphism group of the CPM graphs and investigate isomorphisms between them. Moreover, we determine which of these graphs are 2-arc-transitive, which are arc-transitive but not 2-arc-transitive, and which are half-arc-transitive.

The Purity Conjecture in type C

A collection 𝒞 of k-element subsets of {1,2,...,m} is weakly separated if for each I,J∈𝒞, when the integers 1,2,...,m are arranged around a circle, there is a chord separating I∖J from J∖I. Oh, Postnikov and Speyer constructed a correspondence between weakly separated collections which are maximal by inclusion and reduced plabic graphs, a class of networks defined by Postnikov which give coordinate charts on the Grassmannian of k-planes in m-space. As a corollary, they proved Scott’s Purity Conjecture, which states that a weakly separated collection is maximal by inclusion if and only if it is maximal by size. In this note, we describe maximal weakly separated collections corresponding to symmetric plabic graphs, which give coordinate charts on the Lagrangian Grassmannian, and prove a symmetric version of the Purity Conjecture.

Classifying Heptavalent Symmetric Graphs of Order 40p

Classifying Heptavalent Symmetric Graphs of Order 40p

The smallest symmetric cubic graphs with given type

It is known that arc-transitive group actions on finite cubic (3-valent) graphs fall into seven classes, denoted by 1, 21, 22, 3, 41, 42 and 5, where k, k1 or k2 indicates that the action is k-arc-regular, with k2 indicating that there is no arc-reversing automorphism of order 2 (for k=2 or 4). These classes can be further subdivided into 17 sub-classes, according to the types of arc-transitive subgroups of the full automorphism group of the graph, sometimes called the ‘action type’ of the graph. In this paper, we complete the determination of the smallest graphs in each of these 17 classes (begun by Conder and Nedela in (2009) [7]).

A bidirectional graph neural network for traveling salesman problems on arbitrary symmetric graphs

Deep learning has recently been shown to provide great achievement to the traveling salesman problem (TSP) on the Euclidean graphs. These methods usually fully represent the graph by a set of coordinates, and then captures graph information from the coordinates to generate the solution. The TSP on arbitrary symmetric graphs models more realistic applications where the working graphs maybe sparse, or the distance between points on the graphs may not satisfy the triangle inequality. When prior learning-based methods being applied to the TSP on arbitrary symmetric graphs, they are not capable to capture graph features that are beneficial to produce near-optimal solutions. Moreover, they suffer from serious exploration problems. This paper proposes a bidirectional graph neural network (BGNN) for the arbitrary symmetric TSP. The network learns to produce the next city to visit sequentially by imitation learning. The bidirectional message passing layer is designed as the most important component of BGNN. It is able to encode graphs based on edges and partial solutions. By this way, the proposed approach is much possible to construct near-optimal solutions for the TSP on arbitrary symmetric graphs, and it is able to be combined with informed search to further improve performance.

Rapid Transitions With Robust Accelerated Delayed-Self-Reinforcement for Consensus-Based Networks

Rapid transitions are important for quick response of consensus-based, multi-agent networks to external stimuli. While high-gain can increase response speed, potential instability tends to limit the maximum possible gain, and therefore, limits the maximum convergence rate to consensus during transitions. Since the update law for multi-agent networks with symmetric graphs can be considered as the gradient of its Laplacian-potential function, Nesterov-type accelerated-gradient approaches from optimization theory, can further improve the convergence rate of such networks. An advantage of the accelerated-gradient approach is that it can be implemented using accelerated delayed-self-reinforcement (A-DSR), which does not require new information from the network nor modifications in the network connectivity. However, the accelerated-gradient approach is not directly applicable to general directed graphs since the update law is not the gradient of the Laplacian-potential function. The main contribution of this work is to extend the accelerated-gradient approach to general directed graph networks, without requiring the graph to be strongly connected. Additionally, while both the momentum term and outdated-feedback term in the accelerated-gradient approach are important in general, it is shown that the momentum term alone is sufficient to achieve balanced robustness and rapid transitions without oscillations in the dominant mode, for networks whose graph Laplacians have real spectrum. Simulation results are presented to illustrate the performance improvement with the proposed Robust A-DSR of 40% in structural robustness and 50% in convergence rate to consensus, when compared to the case without the A-DSR. Moreover, experimental results are presented that show a similar 37% faster convergence with the Robust A-DSR when compared to the case without the A-DSR.

Analysis of lackadaisical quantum walks

The lackadaisical quantum walk is a quantum analogue of the lazy random walk obtained by adding a self-loop to each vertex in the graph. We analytically prove that lackadaisical quantum walks can find a unique marked vertex on any regular locally arc-transitive graph with constant success probability quadratically faster than the hitting time. This result proves several speculations and numerical findings in previous work, including the conjectures that the lackadaisical quantum walk finds a unique marked vertex with constant success probability on the torus, cycle, Johnson graphs, and other classes of vertex-transitive graphs. Our proof establishes and uses a relationship between lackadaisical quantum walks and quantum interpolated walks for any regular locally arc-transitive graph.

Ексерго-економічна оптимізація комплексних систем енергопостачання

The optimization of energy supply system becomes especially important problem in those cases where there are several different energy sources, including, e.g., renewable energy sources, and several energy sinks of different power. This problems can be solved with the use of a graph of exergy and economic expenditures for the pairwise interaction of flows. The purpose of the study is to specify the concept of exergy schedule and economic costs applied to energy supply systems (ESS). We shall interpret a graph of the exergy and economic expenditures of an ESS, having an arbitrary structure, as a bipartite graph Z such that the set of its nodes C corresponds to the heating H and heated C flows, and the set of its arcs D to a possible distribution of the exergy and economic expenditures in the corresponding elements of this ESS in the course of interaction between the heating and heated flows. A symmetric graph represents an oriented graph, whose arcs can be grouped into pairs of parallel but oppositely directed arcs. Such graphs, having no isolated nodes, are convenient for studying complex interrelated systems. If we have determined the optimal pair of elements (аі, aj), corresponding to the sequence of nodes, beginning from the root of the foretree and finishing by a suspended node, giving a matrix of unit dimension, then the obtained sequence of elements forms a single-contour system with the minimum level of exergy and economic expenditure. For finding the optimal solution it is advisable to use the method of branches and boundaries, which enables one to improve the solution simpler than with the application of different methods of exergy analysis.

A note on automorphism groups of symmetric cubic graphs

We study [Formula: see text]-arc-transitive cubic graph [Formula: see text], and give a characterization of minimal normal subgroups of the automorphism group. In particular, each [Formula: see text] with quasi-primitive automorphism group is characterized. An interesting consequence of this characterization states that a non-solvable minimal normal subgroup [Formula: see text] contains at most 2 copies of non-abelian simple group when it acts transitively on arcs, or contains at most 6 copies of non-abelian simple group when it acts regularly on vertices.

Gathering robots in graphs: The central role of synchronicity

The Gathering task for k robots disposed on the n vertices of a graph G requires robots to move toward a common vertex from where they do not move anymore.When dealing with very weak robots in terms of capabilities, considering synchronous or asynchronous settings may heavily affect the feasibility of the problem. In fact, even though dealing with asynchronous robots in general requires more sophisticated strategies with respect to the synchronous counterpart, sometimes it comes out that asynchronous robots simply cannot solve the problem whereas synchronous robots can.We study general properties of graphs that can be exploited in order to accomplish the gathering task in the synchronous setting, obtaining an interesting and innovative sufficient condition for the feasibility of the gathering task in graphs, regardless the topology.Furthermore, we consider dense and symmetric graphs like complete and complete bipartite graphs where in general the topology does not allow to distinguish vertices where to finalize the gathering. In such topologies, we fully characterize the solvability of the gathering task in the synchronous setting by suitably combining some strategies arising from the general approach with specific techniques dictated by the considered topologies.From the lower bound point of view in terms of number of synchronous time units required to accomplish the gathering task, in general nothing better than Ω(DG), with DG being the diameter of the input graph G, can be provided. For both complete and complete bipartite graphs, we prove a lower bound of Ω(logϕ⁡k), with ϕ being the well-known golden ratio. Combined with the provided algorithms, this reveals to be asymptotically tight for complete graphs while for complete bipartite graphs an additive factor of δ(k) is achieved, with δ(k) being the function that returns the number of divisors of the integer k.

Arc-transitive graphs of valency twice a prime admit a semiregular automorphism

We prove that every finite arc-transitive graph of valency twice a prime admits a nontrivial semiregular automorphism, that is, a non-identity automorphism whose cycles all have the same length. This is a special case of the Polycirculant Conjecture, which states that all finite vertex-transitive digraphs admit such automorphisms.

LDPC codes constructed from cubic symmetric graphs

Low-density parity-check (LDPC) codes have been the subject of much interest due to the fact that they can perform near the Shannon limit. In this paper we present a construction of LDPC codes from cubic symmetric graphs. The codes constructed are (3, 3)-regular and the vast majority of the corresponding Tanner graphs have girth greater than four. We analyse properties of the codes obtained and present bounds for the code parameters, the dimension and the minimum distance. Furthermore, we give an expression for the variance of the syndrome weight of the codes constructed. Information on the LDPC codes constructed from bipartite cubic symmetric graphs with less than 200 vertices is presented as well. Some of the codes constructed are optimal, and some have an additional property of being self-orthogonal or linear codes with complementary dual (LCD codes).

On Reconstruction of Normal Edge-Transitive Cayley Graphs

The main idea of this paper is to provide an algebraic algorithm for constructing symmetric graphs with optimal fault tolerance. For this purpose, we use normal edge-transitive Cayley graphs and the idea of reconstruction question posed by Praeger to present a special factorization of groups which induces a graphical decomposition of normal edge-transitive Cayley graphs to simpler normal edge-transitive Cayley graphs. Then as a consequence of our results, we continue the study of normal edge-transitive Cayley graphs of abelian groups and we show that knowing normal edge-transitive Cayley graphs of abelian p-groups, we can determine all normal edge-transitive Cayley graphs of abelian groups.

Lithospheric mantle anisotropy from local events beneath the Sunda–Banda arc transition and its geodynamic implications

Shear wave splitting analysis to characterise lithospheric mantle anisotropy has been performed to provide better knowledge about lithospheric deformation and mantle flow beneath the Sunda–Banda arc transition, Indonesia. The tectonic setting of the study area is very complex characterised by the transition from subduction along Sunda arc to collision in Banda arc. The splitting measurements show lateral and vertical variation in the fast directions of the S-waves in this region. When the splitting results are analysed through 2D delay-time tomography and spatial averaging, systematic patterns in delay times and fast polarisation become more visible. In the subduction domain, the spatial averages of fast directions are dominated by two distinct fast polarisations: perpendicular and parallel to the plate motion for shallow and deep events, respectively. The results suggest that anisotropy in this area is not only controlled by anisotropic source related to the simple mantle flow model, but also by anisotropic fabric in the mantle deformed under influence of high stresses, high water contents and low temperatures. In addition, there might also be contribution from the anisotropic body in the upper layer. In the collision domain, spatially averaged fast directions show mostly perpendicular to the plate motion for all deep levels. For shallow level in this region, this trend is mainly governed by the lithospheric deformation process due to the continent-arc collision as also shown by delay time tomographic inversion. For deeper part of the region, the result of tomographic inversion and spatial averaging reveals a high anisotropy followed by rotational pattern of fast directions in the north of Timor. We suggest that this pattern might be related to the induced mantle flow due to lateral tearing of the slab.

FUNGSI SIMETRI TERHADAP GARIS x = a DAN SIFAT-SIFATNYA

An even function is a function with a graph that is symmetric with respect to the y-axis or the line x = 0. In this paper, we will introduce a more general function of the even function, we call it as symmetry function with respect to the line x = a, which is a function whose graph is symmetric with respect to the line x = a. This paper discusses the properties of the symmetry function with respect to the line x = a, which is derived from the pre-existing properties of the even function. Some of the results obtained above, the linear combination of the symmetry functions with respect to the line x = a is a symmetry function with respect to the line x = a and the composition of any function with a symmety function with respect to the line x = a is a symmetry function with respect to the line x = a. Keywords: Even function, a symmetry function with respect to the line x=a, symmetric graph

An Algorithm for Counting the Fixed Point Orbits of an AND-OR Dynamical System with Symmetric Positive Dependency Graph

In this paper we present an algorithm which counts the number of fixed point orbits of an AND-OR dynamical system. We further extend the algorithm in order to list all its fixed point orbits (FPOs) in polynomial time on the number of FPOs of the system.

Rarity of extremal edges in random surfaces and other theoretical applications of cluster algorithms

Motivated by questions on the delocalization of random surfaces, we prove that random surfaces satisfying a Lipschitz constraint rarely develop extremal gradients. Previous proofs of this fact relied on reflection positivity and were thus limited to random surfaces defined on highly symmetric graphs, whereas our argument applies to general graphs. Our proof makes use of a cluster algorithm and reflection transformation for random surfaces of the type introduced by Swendsen–Wang, Wolff and Evertz et al. We discuss the general framework for such cluster algorithms, reviewing several particular cases with emphasis on their use in obtaining theoretical results. Two additional applications are presented: A reflection principle for random surfaces and a proof that pair correlations in the spin $O(n)$ model have monotone densities, strengthening Griffiths’ first inequality for such correlations.

Graphs that contain multiply transitive matchings

Let Γ be a finite, undirected, connected, simple graph. We say that a matching M is a permutablem-matching if M contains m edges and the subgroup of Aut(Γ) that fixes the matching M setwise allows the edges of M to be permuted in any fashion. A matching M is 2-transitive if the setwise stabilizer of M in Aut(Γ) can map any ordered pair of distinct edges of M to any other ordered pair of distinct edges of M. We provide constructions of graphs with a permutable matching; we show that, if Γ is an arc-transitive graph that contains a permutable m-matching for m⩾4, then the degree of Γ is at least m; and, when m is sufficiently large, we characterize the locally primitive, arc-transitive graphs of degree m that contain a permutable m-matching. Finally, we classify the graphs that have a 2-transitive perfect matching and also classify graphs that have a permutable perfect matching.