Symmetric Graphs Research Articles

Nash equilibria in symmetric graph games with partial observation

We investigate a model for representing large multiplayer games, which satisfy strong symmetry properties. This model is made of multiple copies of an arena; each player plays in his own arena, and can partially observe what the other players do. Therefore, this game has partial information and symmetry constraints, which make the computation of Nash equilibria difficult. We show several undecidability results, and for bounded-memory strategies, we precisely characterize the complexity of computing pure Nash equilibria for qualitative objectives in this game model.

Crustal anisotropy along the Sunda-Banda arc transition zone from shear wave splitting measurements

We analyse shear wave splitting derived from the local earthquakes recorded at 13 seismic stations to investigate crustal anisotropy over varied geological regimes of the Sunda-Banda arc transition zone. We determine high-quality splitting measurements for 262 event-station pairs. The orientations of fast polarisation for the stations located in the oceanic regime are generally parallel or sub-parallel to the directions of the principal compressional strain-rate axes with a lack of dependency of delay time δt on increasing depth. The results suggest that anisotropy in this domain is primarily due to the influence of stress induced anisotropy on the upper crust. On the other hand, the average fast polarisations show more scattered for the stations located around Sumba Island and in the collision regime, implying a mix of anisotropy causes. Thus, anisotropy in this region is not only controlled by preferentially aligned cracks due to tectonic stress, but also by preferential mineral alignment and macro-scale faults associated with the regional tectonic deformation. We also perform further analysis to search possible temporal variations of splitting parameters associated with the stress changes excited by large earthquakes. The association between variation in splitting parameters and earthquake activity observed in this study might provide useful information about accumulation of stress before large events, and thus might be considered as an earthquake-forecasting tool in the future.

Arc-transitive abelian regular covering graphs

A lot of attention has been paid recently to the construction of symmetric covers of symmetric graphs. After a new approach given by Conder and the author [Arc-transitive abelian regular covers of cubic graphs, J. Algebra 387 (2013) 215–242], the group of covering transformations can be extended to more general abelian groups rather than cyclic or elementary abelian groups. In this paper, by using the Conder–Ma approach, we investigate the symmetric covers of 4-valent symmetric graphs. As an application, all the arc-transitive abelian regular covers of the 4-valent complete graph [Formula: see text] which can be obtained by lifting the arc-transitive subgroups of automorphisms [Formula: see text] and [Formula: see text] are classified.

A lower bound for the energy of symmetric matrices and graphs

The energy of a symmetric matrix is the sum of the absolute values of its eigenvalues. We introduce a lower bound for the energy of a symmetric matrix partitioned into blocks. This bound is related to the spectrum of its quotient matrix. Furthermore, we study necessary conditions for the equality. Applications to the energy of the generalized composition of a family of arbitrary graphs are obtained. A lower bound for the energy of a graph with a bridge is given. Some computational experiments are presented in order to show that, in some cases, the obtained lower bound is incomparable with the well known lower bound 2m, where m is the number of edges of the graph.

Existence and extendibility of rotationally symmetric graphs with a prescribed higher mean curvature function in Euclidean and Minkowski spaces

In this paper we investigate the existence of rotationally symmetric entire graphs (resp. entire spacelike graphs) with prescribed k-th mean curvature function in Euclidean space Rn+1 (resp. Minkowski spacetime Ln+1). As a previous step, we analyze the associated homogeneous Dirichlet problem on a ball, which is not elliptic for k>1, and then we prove that it is possible to extend the solutions. Moreover, a sufficient condition for uniqueness is given in both cases.

Perfect state transfer by means of discrete-time quantum walk search algorithms on highly symmetric graphs

Perfect state transfer between two marked vertices of a graph by means of discrete-time quantum walk is analyzed. We consider the quantum walk search algorithm with two marked vertices, sender and receiver. It is shown by explicit calculation that for the coined quantum walks on star graph and complete graph with self-loops perfect state transfer between the sender and receiver vertex is achieved for arbitrary number of vertices $N$ in $O(\sqrt{N})$ steps of the walk. Finally, we show that Szegedy's walk with queries on complete graph allows for state transfer with unit fidelity in the limit of large $N$.

Bipartite separability and nonlocal quantum operations on graphs

In this paper we consider the separability problem for bipartite quantum states arising from graphs. Earlier it was proved that the degree criterion is the graph-theoretic counterpart of the familiar positive partial transpose criterion for separability, although there are entangled states with positive partial transpose for which the degree criterion fails. Here we introduce the concept of partially symmetric graphs and degree symmetric graphs by using the well-known concept of partial transposition of a graph and degree criteria, respectively. Thus, we provide classes of bipartite separable states of dimension $m \times n$ arising from partially symmetric graphs. We identify partially asymmetric graphs that lack the property of partial symmetry. We develop a combinatorial procedure to create a partially asymmetric graph from a given partially symmetric graph. We show that this combinatorial operation can act as an entanglement generator for mixed states arising from partially symmetric graphs.

Arc-transitive graphs of square-free order and small valency

This paper is one of a series of papers devoted to characterizing edge-transitive graphs of square-free order. It presents a complete list of locally-primitive arc-transitive graphs of square-free order and valency d∈{5,6,7}.

Pentavalent symmetric graphs of order twice a prime power

A connected symmetric graph of prime valency is basic if its automorphism group contains no nontrivial normal subgroup having more than two orbits. Let p be a prime and n a positive integer. In this paper, we investigate properties of connected pentavalent symmetric graphs of order 2pn, and it is shown that a connected pentavalent symmetric graph of order 2pn is basic if and only if it is either a graph of order 6, 16, 250, or a graph of three infinite families of Cayley graphs on generalized dihedral groups—one family has order 2p with p=5 or 5∣(p−1), one family has order 2p2 with 5∣(p±1), and the other family has order 2p4. Furthermore, the automorphism groups of these basic graphs are computed. Similar works on cubic and tetravalent symmetric graphs of order 2pn have been done.It is shown that basic graphs of connected pentavalent symmetric graphs of order 2pn are symmetric elementary abelian covers of the dipole Dip5, and with covering techniques, uniqueness and automorphism groups of these basic graphs are determined. Moreover, symmetric Zpn-covers of the dipole Dip5 are classified. As a byproduct, connected pentavalent symmetric graphs of order 2p2 are classified.

Arc-Transitive Pentavalent Graphs of Square-Free Order

Hua et al. (Discrete Math 311, 2259---2267, 2011) and Yang et al. (Discrete Math. 339, 522---532, 2016) classify arc-transitive pentavalent graphs of order 2pq and of order 2pqr (with p, q, r distinct odd primes), respectively. In this paper, we extend their results by giving a classification of arc-transitive pentavalent graphs of any square-free order.

A Family of Symmetric Graphs with Complete Quotients

A finite graph $\Gamma$ is $G$-symmetric if it admits $G$ as a group of automorphisms acting transitively on $V(\Gamma)$ and transitively on the set of ordered pairs of adjacent vertices of $\Gamma$. If $V(\Gamma)$ admits a nontrivial $G$-invariant partition ${\cal B}$ such that for blocks $B, C \in {\cal B}$ adjacent in the quotient graph $\Gamma_{{\cal B}}$ relative to ${\cal B}$, exactly one vertex of $B$ has no neighbour in $C$, then we say that $\Gamma$ is an almost multicover of $\Gamma_{{\cal B}}$. In this case there arises a natural incidence structure ${\cal D}(\Gamma, {\cal B})$ with point set ${\cal B}$. If in addition $\Gamma_{{\cal B}}$ is a complete graph, then ${\cal D}(\Gamma, {\cal B})$ is a $(G, 2)$-point-transitive and $G$-block-transitive $2$-$(|{\cal B}|, m+1, \lambda)$ design for some $m \geq 1$, and moreover either $\lambda=1$ or $\lambda=m+1$. In this paper we classify such graphs in the case when $\lambda = m+1$; this together with earlier classifications when $\lambda = 1$ gives a complete classification of almost multicovers of complete graphs.

La preuve de la conjecture d’Ulam.

I give in this article one proof to the famous conjecture of Ulam about the (-1)-reconstruction of the symmetrical graphs. .

Locally 3-Arc-Transitive Regular Covers of Complete Bipartite Graphs

In this paper, locally $3$-arc-transitive regular covers of complete bipartite graphs are studied, and results are obtained that apply to arbitrary covering transformation groups. In particular, methods are obtained for classifying the locally $3$-arc-transitive graphs with a prescribed covering transformation group, and these results are applied to classify the locally $3$-arc-transitive regular covers of complete bipartite graphs with covering transformation group isomorphic to a cyclic group or an elementary abelian group of order $p^2$.

Distributed stopping for average consensus in undirected graphs via event-triggered strategies

We develop and analyze two distributed event-triggered linear iterative algorithms that enable the components of a distributed system, each with some initial value, to reach approximate average consensus on their initial values, after executing a finite number of iterations. Each proposed algorithm provides a criterion that allows the nodes to determine, in a distributed manner, when to terminate because approximate average consensus has been reached, i.e., all nodes have obtained a value that is within a small distance from the average of their initial values. We focus on a distributed system whose underlying topology is captured by an undirected (symmetric) graph, and develop linear iterative strategies with time-varying weights, chosen based on the subset of edges that separate nodes with significantly different values and are considered active at each iteration. In our simulations, we illustrate the proposed algorithms and compare the number of iterations and transmitted values required by the proposed protocols against a previously proposed stopping protocol for approximate average consensus.

Pentavalent symmetric graphs of order 2p 3

A graph is symmetric or 1-regular if its automorphism group is transitive or regular on the arc set of the graph, respectively. We classify the connected pentavalent symmetric graphs of order 2p 3 for each prime p. All those symmetric graphs appear as normal Cayley graphs on some groups of order 2p 3 and their automorphism groups are determined. For p = 3, no connected pentavalent symmetric graphs of order 2p 3 exist. However, for p = 2 or 5, such symmetric graph exists uniquely in each case. For p ≥ 7, the connected pentavalent symmetric graphs of order 2p 3 are all regular covers of the dipole Dip5 with covering transposition groups of order p 3, and they consist of seven infinite families; six of them are 1-regular and exist if and only if 5 | (p-1), while the other one is 1-transitive but not 1-regular and exists if and only if 5 | (p± 1). In the seven infinite families, each graph is unique for a given order.

Entire radially symmetric graphs with prescribed mean curvature in warped product spaces

This study investigates entire radially symmetric graphs in the warped product M×fR, where M is a Riemannian manifold with a pole and f:M→R+ is the warping function. The main results demonstrate the existence and uniqueness of entire radially symmetric graphs with prescribed mean curvature.

Locally 3‐Transitive Graphs of Girth 4

AbstractA natural topic of algebraic graph theory is the study of vertex transitive graphs. In the present article, we investigate locally 3‐transitive graphs of girth 4. Taking our former results on locally symmetric graphs of girth 4 as a starting point, we show what properties are retained if we weaken the requirement of local symmetry to local 3‐transitivity.

Experimental investigation of current-induced local scour around composite bucket foundation in silty sand

In recent years, local scour has received attention because it may have a negative effect on coastal structures. Because of the large-scale arc transition and other special constructs, flow patterns around composite bucket foundation are complicated, and the local scour around composite bucket foundation has rarely been reported in the literature. In the present study, a detailed laboratory testing program on model with diameters of 150cm and 75cm embedded in silty sand was conducted in a flume. The scouring process and equilibrium scour hole geometry under steady and bidirectional current were investigated in detail. A procedure was suggested to predict the ultimate scour depth based on the observed variation of the scour depth over a limited time period. The equilibrium scour depth under a bidirectional current was approximately 16% less than that under a steady current, and there were two spoon-shaped scour holes in the steady current scour, whereas four saddle-shaped scour holes were discovered in bidirectional current scour. Based on these results, a functional relationship was suggested between the scour depth and other parameters, such as the diameter of the foundation, Shields parameter and median diameter of the soil bed.

On the distribution of energy of localized solutions of the Schrödinger equation that propagate along symmetric quantum graphs

From the point of view of applications to quantum mechanics, it is natural to pose a question concerning the distribution of energy of localized solutions of a nonstationary Schrodinger equation over the graph (in other words, the probability to find a quantum particle in a given area). This problem is apparently very complicated for general graphs, because the energy distribution is much more sensitive to the form of boundary conditions and to the initial state than the asymptotic behavior of the number of localized functions. Below, we present initial results concerning the distribution of energy in the case of symmetric quantum graphs (this means that the Schrodinger operators on different edges have the same structure). For general local self-adjoint boundary conditions, we describe the process of onestep scattering of the localized solutions and obtain a simple general result of the distribution of energy. Some special cases and specific examples are discussed.

Radially Symmetric Solutions to the Graphic Willmore Surface Equation

We show that a smooth radially symmetric solution u to the graphic Willmore surface equation is either a constant or the defining function of a half sphere in $${\mathbb R}^3$$ . In particular, radially symmetric entire Willmore graphs in $${\mathbb R}^3$$ must be flat. When u is a smooth radial solution over a punctured disk $$D(\rho )\backslash \{0\}$$ and is in $$C^1(D(\rho ))$$ , we show that there exist a constant $$\lambda $$ and a function $$\beta $$ in $$C^0(D(\rho ))$$ such that $$u''(r) =\frac{\lambda }{2}\log r+\beta (r)$$ ; moreover, the graph of u is contained in a graphical region of an inverted catenoid which is uniquely determined by $$\lambda $$ and $$\beta (0)$$ . It is also shown that a radial solution on the punctured disk extends to a $$C^1$$ function on the disk when the mean curvature is square integrable.