Symmetric Graphs Research Articles

Bandwidth-efficient network monitoring algorithms based on segment routing

We study the bandwidth-efficient network monitoring algorithms using segment routing on topologies that can be modelled as a symmetric directed graph. Aiming at directly minimizing the cycle cover length, four new two-phase algorithms, A, B, C and D, are proposed. Algorithm A is a simple extension of an existing algorithm. Algorithm B utilizes the notion of symmetric cycles. Algorithm C enhances B by adopting a new path metric. Algorithm D further improves C with a parallel and sequential cycle construction process. Since two-phase algorithms require a dedicated monitoring topology, the four two-phase algorithms are then enhanced to function without monitoring topology. Although the cycle cover length found is slightly increased, we argue that the cost saving in removing monitoring topology is far more significant.

Invariant Graph Partition Comparison Measures

Symmetric graphs have non-trivial automorphism groups. This article starts with the proof that all partition comparison measures we have found in the literature fail on symmetric graphs, because they are not invariant with regard to the graph automorphisms. By the construction of a pseudometric space of equivalence classes of permutations and with Hausdorff’s and von Neumann’s methods of constructing invariant measures on the space of equivalence classes, we design three different families of invariant measures, and we present two types of invariance proofs. Last, but not least, we provide algorithms for computing invariant partition comparison measures as pseudometrics on the partition space. When combining an invariant partition comparison measure with its classical counterpart, the decomposition of the measure into a structural difference and a difference contributed by the group automorphism is derived.

Characterization of perfect matching transitive graphs

A graph G is perfect matching transitive, shortly PM-transitive, if for any two perfect matchings M and N of G , there is an automorphism f : V ( G ) ↦ V ( G ) such that f e ( M ) = N , where f e ( u v ) = f ( u ) f ( v ) . In this paper, the author proposed the definition of PM-transitive, verified PM-transitivity of some symmetric graphs, constructed several families of PM-transitive graphs which are neither vertex-transitive nor edge-transitive, and discussed PM-transitivity of generalized Petersen graphs.

A class of highly symmetric graphs, symmetric cylindrical constructions and their spectra

In this article, we introduce the algebra of block-symmetric cylinders and we show that symmetric cylindrical constructions on base-graphs admitting commutative decompositions behave as generalized tensor products. We compute the characteristic polynomial of such symmetric cylindrical constructions in terms of the spectra of the base-graph and the cylinders in a general setting. This gives rise to a simultaneous generalization of some well-known results on the spectra of a variety of graph amalgams, as various graph products, graph subdivisions and generalized Petersen graph constructions. While our main result introduces a connection between spectral graph theory and commutative decompositions of graphs, we focus on commutative cyclic decompositions of complete graphs and tree-cylinders along with a subtle group labeling of trees to introduce a class of highly symmetric graphs containing the Petersen and the Coxeter graphs. Also, using techniques based on recursive polynomials we compute the characteristic polynomials of these highly symmetric graphs as an application of our main result.

On Metric Dimensions of Symmetric Graphs Obtained by Rooted Product

Let G = (V, E) be a connected graph and d(x, y) be the distance between the vertices x and y in G. A set of vertices W resolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in W. A metric dimension of G is the minimum cardinality of a resolving set of G and is denoted by dim(G). In this paper, Cycle, Path, Harary graphs and their rooted product as well as their connectivity are studied and their metric dimension is calculated. It is proven that metric dimension of some graphs is unbounded while the other graphs are constant, having three or four dimensions in certain cases.

Quotient-Complete Arc-Transitive Latin Square Graphs from Groups

We consider latin square graphs $$\varGamma = \text {LSG}(H)$$ of the Cayley table of a given finite group H. We characterize all pairs $$(\varGamma ,G)$$ , where G is a subgroup of autoparatopisms of the Cayley table of H such that G acts arc-transitively on $$\varGamma $$ and all nontrivial G-normal quotient graphs of $$\varGamma $$ are complete. We show that H must be elementary abelian and determine the number k of complete normal quotients. This yields new infinite families of diameter two arc-transitive graphs with $$k = 1$$ or $$k = 2$$ .

Pentavalent symmetric graphs of order four times an odd square-free integer

A graph is said to be symmetric if its automorphism group is transitive on its arcs. Guo et al. in 2011 and Pan et al. in 2013 determined all pentavalent symmetric graphs of order 4 p q . In this paper, we shall generalize this result by determining all connected pentavalent symmetric graphs of order four times an odd square-free integer. It is shown in this paper that, for each such graph Γ , either the full automorphism group A u t Γ is isomorphic to P S L (2, p ) , P G L (2, p ) , P S L (2, p ) × ℤ 2 or P G L (2, p ) × ℤ 2 , or Γ is isomorphic to one of 9 graphs.

Prime-valent arc-transitive basic graphs with order 4 p or 4 p 2

Prime-valent arc-transitive basic graphs with order 4 p or 4 p 2

Arc-transitive cyclic and dihedral covers of pentavalent symmetric graphs of order twice a prime

A regular cover of a connected graph is called cyclic or dihedral if its transformation group is cyclic or dihedral respectively, and arc-transitive (or symmetric ) if the fibre-preserving automorphism subgroup acts arc-transitively on the regular cover. In this paper, we give a classification of arc-transitive cyclic and dihedral covers of a connected pentavalent symmetric graph of order twice a prime. All those covers are explicitly constructed as Cayley graphs on some groups, and their full automorphism groups are determined.

Controlled quantum search

Quantum searching for one of N marked items in an unsorted database of n items is solved in $$\mathcal {O}(\sqrt{n/N})$$ steps using Grover’s algorithm. Using nonlinear quantum dynamics with a Gross–Pitaevskii-type quadratic nonlinearity, Childs and Young (Phys Rev A 93:022314, 2016, https://doi.org/10.1103/PhysRevA.93.022314 ) discovered an unstructured quantum search algorithm with a complexity $$\mathcal {O}( \min \{ 1/g \, \log (g n), \sqrt{n} \}) $$ , which can be used to find a marked item after $$o(\log (n))$$ repetitions, where g is the nonlinearity strength. In this work we develop an quantum search on a complete graph using a time-dependent nonlinearity which obtains one of the N marked items with certainty. The protocol has runtime $$\mathcal {O}(n /(g \sqrt{N(n-N)}))$$ , where g is related to the time-dependent nonlinearity. We also extend the analysis to a quantum search on general symmetric graphs and can greatly simplify the resulting equations when the graph diameter is less than 4.

Evaluating aesthetics for user-sketched layouts of symmetric graphs

In evaluating aesthetics of graph drawings, recent works have provided participants with an adjacency list of a graph and asked them to sketch a pleasing layout of the graph. However, this line of research has not considered aesthetics for user-sketched layouts of graphs with symmetries, which are an important property of a given graph drawing. Therefore, the work reported in this paper focuses on empirically evaluating aesthetics for user-sketched layouts of symmetric graphs. We provided participants adjacency lists of two symmetric graphs and asked them to sketch the two graphs. We analyzed whether symmetries were shown in user-sketched drawings, and further analyzed the aesthetics based on their drawing processes and drawing preferences. In addition, our experiment includes multiple stages to analyze whether the participants had the potential learning ability to draw symmetric structures of graphs. We found that in drawing symmetrical graphs, showing symmetry was likely considered more important than minimizing edge crossings, and that most participants made substantial progress in their ability to create symmetric drawings after going through the experimental stages.

On Arc-Transitive Metacyclic Covers of Graphs with Order Twice a Prime

Quite a lot of attention has been paid recently to the characterization and construction of edge- or arc-transitive abelian (mostly cyclic or elementary abelian) covers of symmetric graphs, but there are rare results for nonabelian covers since the voltage graph techniques are generally not easy to be used in this case. In this paper, we will classify certain metacyclic arc-transitive covers of all non-complete symmetric graphs with prime valency and twice a prime order $2p$ (involving the complete bipartite graph ${\sf K}_{p,p}$, the Petersen graph, the Heawood graph, the Hadamard design on $22$ points and an infinite family of prime-valent arc-regular graphs of dihedral groups). A few previous results are extended.

Analysis and discussion on launching mechanism and tactical electromagnetic railgun technology

Abstract This paper begins with a discussion on the significance of multi-physical fields in the research of railgun launching mechanism, followed by an introduction of research work about dynamic measurements and numerical simulations. The application of some measurement methods including the atomic emission spectrum, fiber-optic strain, optical level, pulsed X-ray, and high-speed video in observing the launching process was introduced. The models about the electromechanical dynamic processes and multi-physical fields in a railgun were developed. The mechanisms of the grooving, arc transition, and gouging were analyzed. These work provided a deeper understanding of launching mechanism of electromagnetic railgun. Some issues about the tactical railgun, such as the fiber composite overwrapped barrel, recoil, loading modes and eddy current loss were also discussed. Taking a medium-caliber launcher as an example, the basic process of design and analysis was introduced.

Sectoriality and Essential Spectrum of Non Symmetric Graph Laplacians

We consider a non self-adjoint Laplacian on a directed graph with non symmetric edge weights. We give sufficient conditions for this Laplacian to be sectorial. We introduce a special self-adjoint operator and compare its essential spectrum with that of the non self-adjoint Laplacian considered.

Multipartite Separability of Density Matrices of Graphs

A new layers method is presented for multipartite separability of density matrices from simple graphs. Full separability of tripartite states is studied for graphs on degree symmetric premise. The models are generalized to multipartite systems by presenting a class of fully separable states arising from partially symmetric graphs.

English

A graph is said to be symmetric if its automorphism group acts transitively on its arcs. In this paper, all connected valency seven symmetric graphs of order 2pq are classified, where p, q are distinct primes. It follows from the classification that there is a unique connected valency seven symmetric graph of order 4p, and that for odd primes p and q, there is an infinite family of connected valency seven one-regular graphs of order 2pq with solvable automorphism groups, and there are four sporadic ones with nonsolvable automorphism groups, which is 1, 2, 3-arc transitive, respectively. In particular, one of the four sporadic ones is primitive, and the other two of the four sporadic ones are bi-primitive.

Belief propagation on replica symmetric random factor graph models

According to physics predictions, the free energy of random factor graph models that satisfy a certain “static replica symmetry” condition can be calculated via the Belief Propagation message passing scheme [Krzakala et al., PNAS 2007]. Here we prove this conjecture for two general classes of random factor graph models, namely Poisson random factor graphs and random regular factor graphs. Specifically, we show that the messages constructed just as in the case of acyclic factor graphs asymptotically satisfy the Belief Propagation equations and that the free energy density is given by the Bethe free energy formula.

Pass-Band Characteristics of an L-Shaped Waveguide in a Diamond Structure Photonic Crystal

The conduction characteristics of a L-shaped waveguide in a diamond structure photonic crystal is investigated in this paper. The waveguides were fabricated with titanium dioxide ceramic via 3-D printing and sintering. The effects of the position and size of line defects on the transmission characteristics are first simulated using a finite-difference time-domain method. The simulated results show that, when the length of the rectangular defect equals the lattice constant, multiple extended modes are generated. When the centers of the single unit cell of the diamond structure and the line defect waveguide coincide, higher transmission efficiency in the line defect can be achieved. In addition, the corner of the L-shaped waveguide was optimized to reduce reflection loss at the turning point using the arc transition of the large diameter. Our experimental results indicate that L-shaped waveguides with an optimized photonic band gap structure and high-K materials can produce a pass-band between 13.8 GHz and 14.4 GHz and increase transmission efficiency. The computed results agree with the experimental results. Our results may help the integration of microwave devices in the future and possibly enable new applications of photonic crystals.

On Pentavalent Arc-Transitive Graphs

In this paper, a characterization of all pentavalent arc-transitive graphs is given. It is shown that each pentavalent arc-transitive covering graph Γ is a regular simple or elementary abelian covering graph. In particular, the elementary abelian covering groups are ℤ3, ℤ5 or a subgroup of [Formula: see text].

Locally-primitive Arc-transitive 10-Valent Graphs of Square-free Order

In this paper, we present a complete classification for locally-primitive arc-transitive graphs which have square-free order and valency 10. The classification involves nine graphs and three infinite families of graphs.