Graph Symmetry Research Articles

The separated box product of two digraphs

A new product construction of graphs and digraphs, called the separated box product, is presented, and several of its properties are discussed. The construction is based on the standard box product of digraphs. However, unlike the standard box product, it preserves the valence of the factor digraphs: If every vertex in both factors has in-valence and out-valence k for some fixed k, then so does every vertex of the separated box product. Questions about the symmetries of the product and their relations to symmetries of the factor graphs are considered. An application of this construction to the case of tetravalent edge-transitive graphs is discussed in detail.

Graph fibrations and symmetries of network dynamics

Dynamical systems with a network structure can display remarkable phenomena such as synchronisation and anomalous synchrony breaking. A methodology for classifying patterns of synchrony in networks was developed by Golubitsky and Stewart. They showed that the robustly synchronous dynamics of a network is determined by its quotient networks. This result was recently reformulated by DeVille and Lerman, who pointed out that the reduction from a network to a quotient is an example of a graph fibration. The current paper exploits this observation and demonstrates the importance of self-fibrations of network graphs. Self-fibrations give rise to symmetries in the dynamics of a network. We show that every network admits a lift with a semigroup or semigroupoid of self-fibrations. The resulting symmetries impact the global dynamics of the network and can therefore be used to explain and predict generic scenarios for synchrony breaking. Also, when the network has a trivial symmetry groupoid, then every robust synchrony in the lift is determined by symmetry. We finish this paper with a discussion of networks with interior symmetries and nonhomogeneous networks.

Combination of distance and symmetry in some molecular graphs

Suppose G is a connected graph or a union of connected graphs and Γ is a subgroup of Aut(G). The modified Wiener index of G with respect to Γ can be defined as follows:W^Γ(G)=|V(G)|2|Γ|∑u∈V(G)∑g∈Γd(u,g(u)).In this article, this graph invariant for the cycle Cn with respect to all subgroups of Aut(Cn) is computed. As consequences, the modified Wiener indices of some molecular graphs like (3, 6)- and (5, 6)-fullerenes with respect to a subgroup of their symmetry groups are computed. Some open questions are also presented.

Search algorithm of structure anomalies in complete graph based on scattering quantum walk

Quantum walks have been proven to be a useful framework in designing new quantum algorithms, of which the search algorithm is the most notable. Besides a general search for a special vertex, recent researches have shown that quantum walks can also be used to find structural anomalies. Suppose a vertex of complete graph KN is attached to a second graph G, then the kind of structure anomaly will break the symmetry of the complete graph. The search algorithm based on scattering quantum walk model is presented to speed up locating this kind of structure anomaly. The concepts of scattering quantum walk model and collapsed graphs are presented. The definition of the evolutionary operator, which is different from that of a general search, is given. Based on the specific definition of evolutionary operator and the obvious symmetry of complete graph, it is shown that the search space is confined to a low-dimensional collapsed space, and the initial state is chosen to lie in this subspace. To illustrate the evolutionary process of the search algorithm, an example is given in the case that G is a single vertex. Taking advantage of our earlier study on the evolutionary operator of coined quantum walks with Grover coin, calculations of the unitary operator in the collapsed space are greatly simplified. To quantify the evolutionary process of the algorithm, we use the matrix perturbation theory involving a perturbative approach to find the eigenvalues and eigenstates. It is the degenerate zeroth-order eigenvalue 0 = 1 that leads to the interesting parts of the Hilbert space. Most of the recent researches of searching the structure anomalies focus on star graph SN with an unspecified graph G attached to one of its external vertices, where the overall graph is divided into two parts by the central vertex. It is shown that quantum speedup will occur if and only if the eigenvalues associated with these two parts in the N limit are the same. In this paper, we find that the collapsed graph of complete graphs can also be divided into two parts by a single collapsed vertex. As these two parts roughly correspond to the initial state and the desired state respectively, the techniques and results in star graphs can be generalized to the collapse graph on complete graph. What is more, under our definition of unitary evolution operator these two parts in the N limit will always share the same eigenvalue, i.e. 0 = 1, no matter what the structure of graph G is. Based on this, we prove that the search algorithm can find the target vertex in O(N) time steps with a success probability of 1-O(1N). That is to say, the quantum search algorithm gains a quadratic speedup over classical counterpart.

Argumentation and graph properties

Argumentation theory is an area of interdisciplinary research that is suitable to characterise several diverse situations of reasoning and judgement in real world practices and challenges. In the discipline of Artificial Intelligence, argumentation is formalised by reasoning models based on building and evaluation of interacting arguments. In this argumentation framework, the semantics of acceptance plays a fundamental role in the argument evaluation process. The determination of accepted arguments under a given semantics (admissible, preferred, stable, etc.) can be a time-consuming and tedious (in number of steps) process. In this work we try to overcome this substantial process by providing a method to compute accepted arguments from an argumentation framework. The principle of this method is to combine mathematical properties (e.g. symmetry, asymmetry, strong connectivity and irreflexivity) of graphs built from the argumentation system to compute sets of accepted arguments. In this work, we combine several graph properties to provide three main propositions; one for identifying accepted arguments under the admissible, preferred semantics and the other to easily identify stable extension. The proofs of the suggested propositions are detailed and this is part of an approach designed to increase collaborative decision-making by improving the effectiveness of reasoning processes.

Modified Wiener index via canonical metric representation, and some fullerene patches

A variation of the classical Wiener index, the modified Wiener index, that was introduced in 1991 by Graovac and Pisanski , takes into account the symmetries of a given graph. In this paper it is proved that the computation of the modified Wiener index of a graph G can be reduced to the computation of the Wiener indices of the appropriately weighted quotient graphs of the canonical metric representation of G . The computation simplifies in the case when G is a partial cube. The method developed is applied to two infinite families of fullerene patches.

On the information leakage of differentially-private mechanisms

Differential privacy aims at protecting the privacy of participants in statistical databases. Roughly, a mechanism satisfies differential privacy if the presence or value of a single individual in the database does not significantly change the likelihood of obtaining a certain answer to any statistical query posed by a data analyst. Differentially-private mechanisms are often oblivious: first the query is processed on the database to produce a true answer, and then this answer is adequately randomized before being reported to the data analyst. Ideally, a mechanism should minimize leakage, i.e., obfuscate as much as possible the link between reported answers and individuals' data, while maximizing utility, i.e., report answers as similar as possible to the true ones. These two goals, however, are in conflict with each other, thus imposing a trade-off between privacy and utility. In this paper we use quantitative information flow principles to analyze leakage and utility in oblivious differentially-private mechanisms. We introduce a technique that exploits graph symmetries of the adjacency relation on databases to derive bounds on the min-entropy leakage of the mechanism. We consider a notion of utility based on identity gain functions, which is closely related to min-entropy leakage, and we derive bounds for it. Finally, given some graph symmetries, we provide a mechanism that maximizes utility while preserving the required level of differential privacy.

Continuous limit of discrete quantum walks

Quantum walks can be defined in two quite distinct ways: discrete-time and continuous-time quantum walks (DTQWs and CTQWs). For classical random walks, there is a natural sense in which continuous-time walks are a limit of discrete-time walks. Quantum mechanically, in the discrete-time case, an additional "coin space" must be appended for the walk to have nontrivial time evolution. Continuous-time quantum walks, however, have no such constraints. This means that there is no completely straightforward way to treat a CTQW as a limit of DTQW, as can be done in the classical case. Various approaches to this problem have been taken in the past. We give a construction for walks on $d$-regular, $d$-colorable graphs when the coin flip operator is Hermitian: from a standard DTQW we construct a family of discrete-time walks with a well-defined continuous-time limit on a related graph. One can think of this limit as a {\it coined} continuous-time walk. We show that these CTQWs share some properties with coined DTQWs. In particular, we look at spatial search by a DTQW over the 2-D torus (a grid with periodic boundary conditions) of size $\sqrt{N}\times\sqrt{N}$, where it was shown \nocite{AAmbainis08} that a coined DTQW can search in time $O(\sqrt{N}\log{N})$, but a standard CTQW \nocite{Childs2004} takes $\Omega(N)$ time to search for a marked element. The continuous limit of the DTQW search over the 2-D torus exhibits the $O(\sqrt{N}\log{N})$ scaling, like the coined walk it is derived from. We also look at the effects of graph symmetry on the limiting walk, and show that the properties are similar to those of the DTQW as shown in \cite{HariKrovi2007}.

The Yamada polynomial of lens spatial graphs

Let [Formula: see text] be a spatial graph in the 3-sphere which covers a graph in the lens space. We introduce a combinatorial representation of [Formula: see text] which reflects the symmetry of the spatial graph. This representation involves a ribbon n-graph and the generator of the center of the n-braid group. We apply this result to study the Yamada polynomial of a class of lens spatial graphs.

Graph Controllability Classes for the Laplacian Leader-Follower Dynamics

In this paper, we consider the problem of obtaining graph-theoretic characterizations of controllability for the Laplacian-based leader-follower dynamics. Our developments rely on the notion of graph controllability classes, namely, the classes of essentially controllable, completely uncontrollable, and conditionally controllable graphs. In addition to the topology of the underlying graph, the controllability classes rely on the specification of the control vectors; our particular focus is on the set of binary control vectors. The choice of binary control vectors is naturally adapted to the Laplacian dynamics, as it captures the case when the controller is unable to distinguish between the followers and, moreover, controllability properties are invariant under binary complements. We prove that the class of essentially controllable graphs is a strict subset of the class of asymmetric graphs and provide numerical results that suggests that the ratio of essentially controllable graphs to asymmetric graphs increases as the number of vertices increases. Although graph symmetries play an important role in graph-theoretic characterizations of controllability, we provide an explicit class of asymmetric graphs that are completely uncontrollable, namely the class of block graphs of Steiner triple systems. We prove that for graphs on four and five vertices, a repeated Laplacian eigenvalue is a necessary condition for complete uncontrollability but, however, show through explicit examples that for eight and nine vertices, a repeated eigenvalue is not necessary for complete uncontrollability. For the case of conditional controllability, we give an easily checkable necessary condition that identifies a class of binary control vectors that result in a two-dimensional controllable subspace. Several constructive examples demonstrate our results.

The SVE Method for Regular Graph Isomorphism Identification

A regular graph is a type of graph that has particular features. Such graphs have great symmetry, which increases the difficulty of isomorphism identification. In this paper, we propose a new method to address regular graph isomorphism identification. We make some adjustments to the original circuit simulation (CS) method in the graph isomorphism identification process, using single-vertex excitation (SVE) in place of the original excitation, which could remove the confusion of corresponding vertices caused by the graph's symmetry. This new SVE method could widen the range of the application of the original CS method and create better performance in time consumption compared with other traditional methods. The performance of the SVE method is provided in this paper illuminating its practicality and superiority.

Projective representations from quantum enhanced graph symmetries

We define re-gaugings and enhanced symmetries for graphs with group labels on their edges. These give rise to interesting projective representations of subgroups of the automorphism groups of the graphs. We furthermore embed this construction into several higher levels of generalization using category theory and show that they are natural in that language. These include projective representations of the re-gauging groupoid and a novel generalization to all symmetries of the graph.

On geodesic transitive graphs

The main purpose of this paper is to investigate relationships between three graph symmetry properties: s-arc transitivity, s-geodesic transitivity, and s-distance transitivity. A well-known result of Weiss tells us that if a graph of valency at least 3 is s-arc transitive then s≤7. We show that for each value of s≤3, there are infinitely many s-arc transitive graphs that are t-geodesic transitive for arbitrarily large values of t. For 4≤s≤7, the geodesic transitive graphs that are s-arc transitive can be explicitly described, and all but two of these graphs are related to classical generalized polygons. Finally, we show that the Paley graphs and the Peisert graphs, which are known to be distance transitive, are almost never 2-geodesic transitive, with just three small exceptions.

Quantum walk-based search and symmetries in graphs

In this paper we present a lemma, which helps us to establish a link between the distribution of success probabilities from quantum walk based search and the symmetries of the underlying graphs. With the aid of the lemma, we identified certain graph structures of which the quantum walk based search provides high success probabilities at the marked vertices. We also observed that many graph structures and their vertices can be classified according to their structural equivalence using the search probabilities provided by quantum walks, although this method cannot resolve all non-equivalent vertices for strongly regular graphs.

On Symmetry of Uniform and Preferential Attachment Graphs

Motivated by the problem of graph structure compression under realistic source models, we study the symmetry behavior of preferential and uniform attachment graphs. These are two dynamic models of network growth in which new nodes attach to a constant number $m$ of existing ones according to some attachment scheme. We prove symmetry results for $m=1$ and $2$, and we conjecture that for $m\geq 3$, both models yield asymmetry with high probability. We provide new empirical evidence in terms of graph defect. We also prove that vertex defects in the uniform attachment model grow at most logarithmically with graph size, then use this to prove a weak asymmetry result for all values of $m$ in the uniform attachment model. Finally, we introduce a natural variation of the two models that incorporates preference of new nodes for nodes of a similar age, and we show that the change introduces symmetry for all values of $m$.

Controllability of undirected graphs

In control theory, networked dynamical systems have a wide range of engineering applications. In a relational graph among followers (F) and leaders (R), new necessary and sufficient conditions for a pair (F,R) to be controllable are presented. The choice of leader vertices for controllability is shown to be facilitated by identifying the core vertices associated with the eigenvectors of a matrix S related to a graph. We present new necessary and sufficient conditions for a graph to be controllable relative to its adjacency matrix or to its signless Laplacian without having to evaluate any eigenspaces, which is the criterion usually employed. The symmetries of the system graph represented by S are also shown to aid in the choice of a potential leader vertex that is able to control the follower subgraph on its own. Moreover, we define k-omnicontrollable graphs for controllability by any k leaders and show that simple 1-omnicontrollable graphs have only two possible automorphism groups.

Every finite group action on a compact 3-manifold preserves infinitely many hyperbolic spatial graphs

We consider symmetries of spatial graphs in compact 3-manifolds described by smooth finite group actions. This paper provides a method for constructing an infinite family of hyperbolic spatial graphs with given symmetry by connecting spatial graph versions of hyperbolic tangles in 3-cells of polyhedral cell decompositions induced from triangulations of the 3-manifolds. This method is applicable also to the case of ideal triangulations.

A shortly connected mesh topology for high performance and energy efficient network-on-chip architectures

Network-on-chip-based communication schemes represent a promising solution to the increasing complexity of system-on-chip problems. In this paper, we propose a new mesh-like topology called the shortly connected mesh technology (ScMesh), which is based on the traditional mesh topology, to exploit the graph symmetry properties of interconnection networks. This proposed topology not only enhances network performance by reducing the network diameter, but also provides a lower area/energy solution for interconnection network scenarios. This study analyzes and compares the performance of ScMesh to some newly improved topologies, including the WK-recursive, extended-butterfly fat tree, and diametrical mesh topologies. The experiment results indicate that ScMesh outperforms the other topologies, with throughput increases of 47.71, 33.45, and 18.64 % as well as latency decreases of 45.71, 35.84, and 14.58 % compared to the extended-butterfly fat tree, WK-recursive and diametrical mesh topologies, respectively. In addition, ScMesh achieves 41.22, 32.23, and 15.01 % lower energy consumption and 38.96, 27.43, and 18.21 % lower area overhead than the extended-butterfly fat tree, WK-recursive, and diametrical mesh topologies, respectively.

Representativity of Cayley maps

The motivating question of this paper is to ask when, given a group A and generating set X for A, one can find a Cayley map CM(A,X,ρ) with representativity at least two (also called a strong embedding of the graph; note that in such an embedding, each face is bounded by a cycle of the graph).We first investigate some general consequences of a Cayley map’s having representativity one, which means that there exist faces that are self-adjacent (and consequently these faces are not bounded by proper cycles of the graph). We find that there exist Cayley graphs of Abelian groups all of whose Cayley maps have representativity one. This indicates that if these graphs have strong embeddings (orientable or non-orientable), then these embeddings cannot be obtained by applying the symmetry of Cayley graphs in the most natural way. This also indicates that the Cycle Double Cover of these graphs, if it exists, cannot be constructed by applying the symmetry of these graphs in the most natural way.In addition, we investigate specific consequences in the case |X|=2 or 3; we provide a few classes of examples of A and X which may be forced to have representativity at least two; and we provide a combinatorial formalism for calculating the representativity of a Cayley map without appealing to the “lifted” embedding.

Characterizing graph symmetries through quantum Jensen-Shannon divergence

In this paper we investigate the connection between quantum walks and graph symmetries. We begin by designing an experiment that allows us to analyze the behavior of the quantum walks on the graph without causing the wave function collapse. To achieve this, we base our analysis on the recently introduced quantum Jensen-Shannon divergence. In particular, we show that the quantum Jensen-Shannon divergence between the evolution of two quantum walks with suitably defined initial states is maximum when the graph presents symmetries. Hence, we assign to each pair of nodes of the graph a value of the divergence, and we average over all pairs of nodes to characterize the degree of symmetry possessed by a graph.