Quotient Graph Research Articles

Epidemic Outbreaks in Networks with Equitable or Almost-Equitable Partitions

We study the diffusion of epidemics on networks that are partitioned into local communities. The gross structure of hierarchical networks of this kind can be described by a quotient graph. The rationale of this approach is that individuals infect those belonging to the same community with higher probability than individuals in other communities. In community models the nodal infection probability is thus expected to depend mainly on the interaction of a few, large interconnected clusters. In this work, we describe the epidemic process as a continuous-time individual-based susceptible-infected-susceptible (SIS) model using a first-order mean-field approximation. A key feature of our model is that the spectral radius of this smaller quotient graph (which only captures the macroscopic structure of the community network) is all we need to know in order to decide whether the overall healthy-state defines a globally asymptotically stable or an unstable equilibrium. Indeed, the spectral radius is related to the epidemic threshold of the system. Moreover we prove that, above the threshold, another steady-state exists that can be computed using a lower-dimensional dynamical system associated with the evolution of the process on the quotient graph. Our investigations are based on the graph-theoretical notion of equitable partition and of its recent and rather flexible generalization, that of almost equitable partition.

Periodic Body-and-Bar Frameworks

Periodic body-and-bar frameworks are abstractions of crystalline structures made of rigid bodies connected by fixed-length bars and subject to the action of a lattice of translations. We give a Maxwell--Laman characterization for minimally rigid periodic body-and-bar frameworks in terms of their quotient graphs. As a consequence we obtain efficient polynomial time algorithms for their recognition based on matroid partition and pebble games.

Infinitesimal Rigidity of Symmetric Bar-Joint Frameworks

We propose new symmetry-adapted rigidity matrices to analyze the infinitesimal rigidity of bar-joint frameworks of arbitrary-dimension with Abelian point group symmetries. These matrices define new symmetry-adapted rigidity matroids on group-labeled quotient graphs. Using these new tools, we establish combinatorial characterizations of infinitesimally rigid two-dimensional bar-joint frameworks whose joints are positioned as generically as possible subject to the symmetry constraints imposed by a reflection, a half-turn, or a threefold rotation in the plane. For bar-joint frameworks which are generic with respect to any other cyclic point group in the plane, we provide a number of necessary conditions for infinitesimal rigidity.

On Strongly Sum Difference Quotient Labeling of One-Point Union of Graphs, Chain and Corona Graphs

Abstract In this paper we study strongly sum difference quotient labeling of some graphs that result from three different constructions. The first construction produces one- point union of graphs. The second construction produces chain graph, i.e., a concatenation of graphs. A chain graph will be strongly sum difference quotient graph if any graph in the chain, accepts strongly sum difference quotient labeling. The third construction is the corona product; strongly sum difference quotient labeling of corona graph is obtained.

Symmetric cubic graphs with solvable automorphism groups

A cubic graph Γ is G-arc-transitive if G≤Aut(Γ) acts transitively on the set of arcs of Γ, and G-basic if Γ is G-arc-transitive and G has no non-trivial normal subgroup with more than two orbits. Let G be a solvable group. In this paper, we first classify all connected G-basic cubic graphs and determine the group structure for every G. Then, combining covering techniques, we prove that a connected cubic G-arc-transitive graph is either a Cayley graph, or its full automorphism group is of type 22, that is, a 2-regular group with no involution reversing an edge, and has a non-abelian normal subgroup such that the corresponding quotient graph is the complete bipartite graph of order 6.

Spectral and asymptotic properties of Grover walks on crystal lattices

We propose a twisted Szegedy walk for estimating the limit behavior of a discrete-time quantum walk on a crystal lattice, an infinite abelian covering graph, whose notion was introduced by [14]. First, we show that the spectrum of the twisted Szegedy walk on the quotient graph can be expressed by mapping the spectrum of a twisted random walk onto the unit circle. Secondly, we show that the spatial Fourier transform of the twisted Szegedy walk on a finite graph with appropriate parameters becomes the Grover walk on its infinite abelian covering graph. Finally, as an application, we show that if the Betti number of the quotient graph is strictly greater than one, then localization is ensured with some appropriated initial state. We also compute the limit density function for the Grover walk on Zd with flip flop shift, which implies the coexistence of linear spreading and localization. We partially obtain the abstractive shape of the limit density function: the support is within the d-dimensional sphere of radius 1/d, and 2d singular points reside on the sphere's surface.

Wreath products by a leavitt path algebra and affinizations

We introduce ring theoretic constructions that are similar to the construction of wreath product of groups [M. Kargapolov and Y. Merzlyakov, Fundamentals of the Theory of Groups (Springer-Verlag, New York, 1979)]. In particular, for a given graph Γ = (V, E) and an associate algebra A, we construct an algebra B = A wr L(Γ) with the following property: B has an ideal I, which consists of (possibly infinite) matrices over A, B/I ≅ L(Γ), the Leavitt path algebra of the graph Γ. Let W ⊂ V be a hereditary saturated subset of the set of vertices [G. Abrams and G. Aranda Pino, The Leavitt path algebra of a graph, J. Algebra 293(2) (2005) 319–334], Γ(W) = (W, E(W, W)) is the restriction of the graph Γ to W, Γ/W is the quotient graph [G. Abrams and G. Aranda Pino, The Leavitt path algebra of a graph, J. Algebra 293(2) (2005) 319–334]. Then L(Γ) ≅ L(W) wr L(Γ/W). As an application we use wreath products to construct new examples of (i) affine algebras with non-nil Jacobson radicals, (ii) affine algebras with non-nilpotent locally nilpotent radicals.

Linking rigid bodies symmetrically

The mathematical theory of rigidity of body–bar and body–hinge frameworks provides a useful tool for analyzing the rigidity and flexibility of many articulated structures appearing in engineering, robotics and biochemistry. In this paper we develop a symmetric extension of this theory which permits a rigidity analysis of body–bar and body–hinge structures with point group symmetries.The infinitesimal rigidity of body–bar frameworks can naturally be formulated in the language of the exterior (or Grassmann) algebra. Using this algebraic formulation, we derive symmetry-adapted rigidity matrices to analyze the infinitesimal rigidity of body–bar frameworks with Abelian point group symmetries in an arbitrary dimension. In particular, from the patterns of these new matrices, we derive combinatorial characterizations of infinitesimally rigid body–bar frameworks which are generic with respect to a point group of the form Z/2Z×⋯×Z/2Z. Our characterizations are given in terms of packings of bases of signed-graphic matroids on quotient graphs. Finally, we also extend our methods and results to body–hinge frameworks with Abelian point group symmetries in an arbitrary dimension.

SYMMETRIC GRAPHS WITH 2-ARC TRANSITIVE QUOTIENTS

AbstractA graph $\Gamma $ is $G$-symmetric if $\Gamma $ admits $G$ as a group of automorphisms acting transitively on the set of vertices and the set of arcs of $\Gamma $, where an arc is an ordered pair of adjacent vertices. In the case when $G$ is imprimitive on $V(\Gamma )$, namely when $V(\Gamma )$ admits a nontrivial $G$-invariant partition ${\mathcal{B}}$, the quotient graph $\Gamma _{\mathcal{B}}$ of $\Gamma $ with respect to ${\mathcal{B}}$ is always $G$-symmetric and sometimes even $(G, 2)$-arc transitive. (A $G$-symmetric graph is $(G, 2)$-arc transitive if $G$ is transitive on the set of oriented paths of length two.) In this paper we obtain necessary conditions for $\Gamma _{{\mathcal{B}}}$ to be $(G, 2)$-arc transitive (regardless of whether $\Gamma $ is $(G, 2)$-arc transitive) in the case when $v-k$ is an odd prime $p$, where $v$ is the block size of ${\mathcal{B}}$ and $k$ is the number of vertices in a block having neighbours in a fixed adjacent block. These conditions are given in terms of $v, k$ and two other parameters with respect to $(\Gamma , {\mathcal{B}})$ together with a certain 2-point transitive block design induced by $(\Gamma , {\mathcal{B}})$. We prove further that if $p=3$ or $5$ then these necessary conditions are essentially sufficient for $\Gamma _{{\mathcal{B}}}$ to be $(G, 2)$-arc transitive.

Non-crystallographic nets: characterization and first steps towards a classification

Non-crystallographic (NC) nets are periodic nets characterized by the existence of non-trivial bounded automorphisms. Such automorphisms cannot be associated with any crystallographic symmetry in realizations of the net by crystal structures. It is shown that bounded automorphisms of finite order form a normal subgroup F(N) of the automorphism group of NC nets (N, T). As a consequence, NC nets are unstable nets (they display vertex collisions in any barycentric representation) and, conversely, stable nets are crystallographic nets. The labelled quotient graphs of NC nets are characterized by the existence of an equivoltage partition (a partition of the vertex set that preserves label vectors over edges between cells). A classification of NC nets is proposed on the basis of (i) their relationship to the crystallographic net with a homeomorphic barycentric representation and (ii) the structure of the subgroup F(N).

Half-arc-transitive group actions with a small number of alternets

A graph X is said to be G-half-arc-transitive if G⩽Aut(X) acts transitively on the set of vertices of X and on the set of edges of X but does not act transitively on the set of arcs of X. Such graphs can be studied via corresponding alternets, that is, equivalence classes of the so-called reachability relation, first introduced by Cameron, Praeger and Wormald (1993) in [5]. If the vertex sets of two adjacent alternets either coincide or have half of their vertices in common the graph is said to be tightly attached. In this paper graphs admitting a half-arc-transitive group action with at most five alternets are considered. In particular, it is shown that if the number of alternets is at most three, then the graph is necessarily tightly attached, but there exist graphs with four and graphs with five alternets which are not tightly attached. The exceptional graphs all admit a partition giving the rose window graph R6(5,4) on 12 vertices as a quotient graph in case of four alternets, and a particular graph on 20 vertices in the case of five alternets.

Computing quaternion quotient graphs via representations of orders

We give a method to describe the quotient of the local Bruhat–Tits tree TP for PGL2(K), where K is a global function field, by certain subgroups of PGL2(K) of arithmetical significance. In particular, we can compute the quotient of TP by an arithmetic subgroup PGL2(A), where A=AP is the ring of functions that are regular outside P, recursively for a place P of any degree, when K is a rational function field. We achieve this by proving that the infinite matrices whose coordinates are the numbers of neighbors of a vertex in TP corresponding to orders in a fixed isomorphism class commute for different places P, using tools from the theory of representations of orders. The latter result holds for every global function field K.

Cut method and Djoković-Winklerʼs relation

Let Θ⁎ be the transitive closure of the Djoković-Winklerʼs relation Θ. It is proved that the Wiener index of a weighted graph (G, w) can be expressed as the sum of the Wiener indices of weighted quotient graphs with respect to an arbitrary combination of Θ⁎-classes. A related result for edge-weighted graphs is also given and a class of graphs studied in [H. Yousefi-Azari, M. H. Khalifeh, and A. R. Ashrafi, Calculating the edge Wiener and Szeged indices of graphs, J. Comput. Appl. Math. 235 (2011), 4866–4870] is characterized as partial cubes. We will Compute distance polynomial functions on graphs with transitive Djoković-Winklerʼs relation.

Strict Inequalities for Connective Constants of Transitive Graphs

The connective constant of a graph is the exponential growth rate of the number of self-avoiding walks starting at a given vertex. Strict inequalities are proved for connective constants of vertex-transitive graphs. First, the connective constant decreases strictly when the graph is replaced by a nontrivial quotient graph. Second, the connective constant increases strictly when a quasi-transitive family of new edges is added. These results have the following implications for Cayley graphs: the connective constant of a Cayley graph decreases strictly when a new relator is added to the group, and increases strictly when a nontrivial group element is declared to be a generator.

The Grid Property and Product-Like Hypergraphs

Equivalence relations on the edge set of a hypergraph that satisfy the "grid property" (a certain restrictive condition on diagonal-free grids that can be seen as a generalization of the more familiar "square property" on graphs) play a crucial role in the theory of Cartesian hypergraph products. In particular, every convex relation with the grid property induces a factorization w.r.t. the Cartesian product. In the class of graphs, even non-convex relations with the square property provide rich structural information on local isomorphisms, local product structures, and product structures of quotient graphs. Here, we examine the grid property in its own right. Vertex partitions derived from these equivalence classes of the edges give rise to equivalence relations on the vertex set. This in turn determines quotient graphs that have non-trivial product structures.

Unique square property, equitable partitions, and product-like graphs

Equivalence relations on the edge set of a graph G that satisfy restrictive conditions on chordless squares play a crucial role in the theory of Cartesian graph products and graph bundles. We show here that such relations in a natural way induce equitable partitions on the vertex set of G, which in turn give rise to quotient graphs that can have a rich product structure even if G itself is prime.

Convenient adjacencies for structuring the digital plane

We study graphs with the vertex set ?2$\mathbb Z^{2}$ which are subgraphs of the 8-adjacency graph and have the property that certain natural cycles in these graphs are Jordan curves, i.e., separate ?2$\mathbb Z^{2}$ into exactly two connected components. Of these graphs, we determine the minimal ones and study their quotient graphs. The results obtained are used to prove digital analogues of the Jordan curve theorem for several graphs on ?2$\mathbb Z^{2}$. Thus, these graphs are shown to provide background structures on the digital plane ?2$\mathbb Z^{2}$ convenient for studying digital images.

Null and non–rainbow colorings of maximal planar graphs

For maximal planar graphs of order n⩾4, we prove that a vertex–coloring containing no rainbow faces uses at most ⌊2n−13⌋ colors, and this is best possible. The main ingredients in the proof are classical homological tools. By considering graphs as topological spaces, we introduce the notion of a null coloring, and prove that for any graph G a maximal null coloring f is such that the quotient graph G/f is a forest.

THE STABILIZERS IN A DRINFELD MODULAR GROUP OF THE VERTICES OF ITS BRUHAT–TITS TREE: AN ELEMENTARY APPROACH

Let K be an algebraic function field of one variable with constant field k and let [Formula: see text] be the Dedekind domain consisting of all those elements of K which are integral outside a fixed place ∞ of K. When k is finite the group [Formula: see text] plays a central role in the theory of Drinfeld modular curves analogous to that played by SL2(ℤ) in the classical theory of modular forms. When k is finite (respectively, infinite) we refer to a group [Formula: see text] as an arithmetic (respectively, non-arithmetic) Drinfeld modular group. Associated with [Formula: see text] is its Bruhat–Tits tree, [Formula: see text]. The structure of the group is derived from that of the quotient graph [Formula: see text]. Using an elementary approach which refers explicitly to matrices we determine the structure of all the vertex stabilizers of [Formula: see text]. This extends results of Serre, Takahashi and the authors. We also determine all possible valencies of the vertices of [Formula: see text] for the important special case where ∞ has degree 1.

Observability and coarse graining of consensus dynamics through the external equitable partition

Using the intrinsic relationship between the external equitable partition (EEP) and the spectral properties of the graph Laplacian, we characterize convergence and observability properties of consensus dynamics on networks. In particular, we establish the relationship between the original consensus dynamics and the associated consensus of the quotient graph under varied initial conditions, and characterize the asymptotic convergence to the synchronization manifold under nonuniform input signals. We also show that the EEP with respect to a node can reveal nodes in the graph with an increased rate of asymptotic convergence to the consensus value, as characterized by the second smallest eigenvalue of the quotient Laplacian. Finally, we show that the quotient graph preserves the observability properties of the full graph and how the inheritance by the quotient graph of particular aspects of the eigenstructure of the full Laplacian underpins the observability and convergence properties of the system.