Quotient Graph Research Articles

Structural transition in interdependent networks with regular interconnections.

Networks are often made up of several layers that exhibit diverse degrees of interdependencies. An interdependent network consists of a set of graphs G that are interconnected through a weighted interconnection matrix B, where the weight of each intergraph link is a non-negative real number p. Various dynamical processes, such as synchronization, cascading failures in power grids, and diffusion processes, are described by the Laplacian matrix Q characterizing the whole system. For the case in which the multilayer graph is a multiplex, where the number of nodes in each layer is the same and the interconnection matrix B=pI, I being the identity matrix, it has been shown that there exists a structural transition at some critical coupling p^{*}. This transition is such that dynamical processes are separated into two regimes: if p>p^{*}, the network acts as a whole; whereas when p<p^{*}, the network operates as if the graphs encoding the layers were isolated. In this paper, we extend and generalize the structural transition threshold p^{*} to a regular interconnection matrix B (constant row and column sum). Specifically, we provide upper and lower bounds for the transition threshold p^{*} in interdependent networks with a regular interconnection matrix B and derive the exact transition threshold for special scenarios using the formalism of quotient graphs. Additionally, we discuss the physical meaning of the transition threshold p^{*} in terms of the minimum cut and show, through a counterexample, that the structural transition does not always exist. Our results are one step forward on the characterization of more realistic multilayer networks and might be relevant for systems that deviate from the topological constraints imposed by multiplex networks.

Basins of Attraction, Commitment Sets, and Phenotypes of Boolean Networks

The attractors of Boolean networks and their basins have been shown to be highly relevant for model validation and predictive modeling, e.g., in systems biology. Yet, there are currently very few tools available that are able to compute and visualize not only attractors but also their basins. In the realm of asynchronous, non-deterministic modeling not only is the repertoire of software even more limited, but also the formal notions for basins of attraction are often lacking. In this setting, the difficulty both for theory and computation arises from the fact that states may be elements of several distinct basins. In this paper, we address this topic by partitioning the state space into sets that are committed to the same attractors. These commitment sets can easily be generalized to sets that are equivalent w.r.t. the long-term behaviors of pre-selected nodes which leads us to the notions of markers and phenotypes which we illustrate in a case study on bladder tumorigenesis. For every concept, we propose equivalent CTL model checking queries and an extension of the state of the art model checking software NuSMV is made available that is capable of computing the respective sets. All notions are fully integrated as three new modules in our Python package PyBoolNet, including functions for visualizing the basins, commitment sets, and phenotypes as quotient graphs and pie charts.

Symmetric graphs of valency five and their basic normal quotients

A graph Γ is symmetric or arc-transitive if its automorphism group Aut(Γ) is transitive on the arc set of the graph, and Γ is basic if Aut(Γ) has no non-trivial normal subgroup N such that the quotient graph ΓN has the same valency as Γ. In this paper, we classify symmetric basic graphs of order 2qpn and valency 5, where q<p are two primes and n is a positive integer. It is shown that such a graph is isomorphic to a family of Cayley graphs on dihedral groups of order 2q with 5|(q−1), the complete graph K6 of order 6, the complete bipartite graph K5,5 of order 10, or one of the nine sporadic coset graphs associated with non-abelian simple groups. As an application, connected pentavalent symmetric graphs of order kpn for some small integers k and n are classified.

Distance-based topological indices of nanosheets, nanotubes and nanotori of $$\hbox {SiO}_2$$ SiO 2

We have computed distance-based topological indices of nanosheets, nanotubes and nanotori of $$\hbox {SiO}_2$$ which find potential applications in drug, food, and cosmetic industry. As topological indices correlate with physico-chemical properties and estimating efficiency of drug deliveries of these species, we compute the topological indices based on their degrees and distances of the associated molecular graphs. We have obtained exact analytical expressions of various topological indices such as the Wiener, vertex-Szeged, edge-Szeged, edge-vertex-Szeged, Padmakar-Ivan, Schultz and Gutman indices of $$\hbox {SiO}_2$$ nanosheet, nanotube and torus using the cut method which involves decomposing a molecular graph by means of the transitive closure property of Djokovic–Winkler relation to smaller strength-weighted quotient graphs.

Stream graphs and link streams for the modeling of interactions over time

Graph theory provides a language for studying the structure of relations, and it is often used to study interactions over time too. However, it poorly captures the intrinsically temporal and structural nature of interactions, which calls for a dedicated formalism. In this paper, we generalize graph concepts to cope with both aspects in a consistent way. We start with elementary concepts like density, clusters, or paths, and derive from them more advanced concepts like cliques, degrees, clustering coefficients, or connected components. We obtain a language to directly deal with interactions over time, similar to the language provided by graphs to deal with relations. This formalism is self-consistent: usual relations between different concepts are preserved. It is also consistent with graph theory: graph concepts are special cases of the ones we introduce. This makes it easy to generalize higher level objects such as quotient graphs, line graphs, k-cores, and centralities. This paper also considers discrete versus continuous time assumptions, instantaneous links, and extensions to more complex 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 $$k = 2$$ .

Vertex collisions in 3-periodic nets of genus 4.

Unstable nets, by definition, display vertex collisions in any barycentric representation, among which are approximate models for the associated crystal structures. This means that different vertex lattices happen to superimpose when every vertex of a periodic net is located at the centre of gravity of its first neighbours. Non-crystallographic nets are known to be unstable, but crystallographic nets can also be unstable and general conditions for instability are not known. Moreover, examples of unstable nets are still scarce. This article presents a systematic analysis of unstable 3-periodic nets of genus 4, satisfying the restrictions that, in a suitable basis, (i) their labelled quotient graph contains a spanning tree with zero voltage and (ii) voltage coordinates belong to the set {-1, 0, 1}. These nets have been defined by a unique circuit of null voltage in the quotient graph. They have been characterized through a shortest path between colliding vertices. The quotient graph and the nature of the net obtained after identification of colliding vertices, if known, are also provided. The complete list of the respective unstable nets, with a detailed description of the results, can be found in the supporting information.

Bad news for chordal partitions

AbstractReed and Seymour [1998] asked whether every graph has a partition into induced connected nonempty bipartite subgraphs such that the quotient graph is chordal. If true, this would have significant ramifications for Hadwiger's Conjecture. We prove that the answer is “no.” In fact, we show that the answer is still “no” for several relaxations of the question.

Surface embeddings of the Klein and the Möbius–Kantor graphs

This paper describes an invariant representation for finite graphs embedded on orientable tori of arbitrary genus, with working examples of embeddings of the Möbius-Kantor graph on the torus, the genus-2 bitorus and the genus-3 tritorus, as well as the two-dimensional, 7-valent Klein graph on the tritorus (and its dual: the 3-valent Klein graph). The genus-2 and -3 embeddings describe quotient graphs of 2- and 3-periodic reticulations of hyperbolic surfaces. This invariant is used to identify infinite nets related to the Möbius-Kantor and 7-valent Klein graphs.

Motions of grid-like reflection frameworks

Combinatorial characterisations are obtained of symmetric and anti-symmetric infinitesimal rigidity for two-dimensional frameworks with reflectional symmetry in the case of norms where the unit ball is a quadrilateral and where the reflection acts freely on the vertex set. At the framework level, these characterisations are given in terms of induced monochrome subgraph decompositions, and at the graph level they are given in terms of sparsity counts and recursive construction sequences for the corresponding signed quotient graphs.

On the structure of isoclinism classes of the non-commuting graphs

In this paper, we introduce the new concept isoclinism of two non-commuting graphs. We describe it with this hope to determine the properties of the graph with large number of vertices and edges more easier by use of its smaller correspondence graph in its isoclinic class. In 1939, Hekster classified the groups by n-isoclinism which was weaker than isomorphism, where n is a positive integer. The abelian groups are in the same class by group-isoclinism, although their intrinsic properties are not the same. The notion of isoclinic groups is the inspiration to define the isoclinism of two graphs. The isoclinism of two graphs is a pair of significant special isomorphism between the quotient graphs of the given graphs. We observe that all complete 3-partite non-commuting graphs are in the same isoclinic class and the non-commuting graph associated to the dihedral group of order 8, \(D_{8}\) is the representative of this class.

Stochastic generation of complex crystal structures combining group and graph theory with application to carbon

A method is introduced to stochastically generate crystal structures with defined structural characteristics. Reasonable quotient graphs for symmetric crystals are constructed using a random strategy combined with space group and graph theory. Our algorithm enables the search for large-size and complex crystal structures with a specified connectivity, such as threefold ${\mathrm{sp}}^{2}$ carbons, fourfold ${\mathrm{sp}}^{3}$ carbons, as well as mixed ${\mathrm{sp}}^{2}\text{\ensuremath{-}}{\mathrm{sp}}^{3}$ carbons. To demonstrate the method, we randomly construct initial structures adhering to space groups from 75 to 230 and a range of lattice constants, and we identify 281 new ${\mathrm{sp}}^{3}$ carbon crystals. First-principles optimization of these structures show that most of them are dynamically and mechanically stable and are energetically comparable to those previously proposed. Some of the new structures can be considered as candidates to explain the experimental cold compression of graphite.

Topological Indices and Their Applications to Circumcised Donut Benzenoid Systems, Kekulenes and Drugs

ABSTRACTThis paper describes a new technique to compute topological indices of donut benzenoids and Kekulenes with applications to drugs by dissecting the original topological network into smaller strength-weighted quotient graphs relative to the transitive closure of the Djokovi–Winkler relation. We have applied this technique to a series of donut benzenoid graphs obtained by circumcising at least two internal hexagons from parent benzenoid graphs. Such donut coronoid graph finds significant applications in emerging chemical materials of importance to synthetic organic chemistry and drug industry. It comprised of donut coronoid structures such as Kekulenes and related circumscribed structures where a number of graph-theoretical based techniques, such as resonance theory and Clar’s sextets. In this work, we have computed a number of topological indices of these donut coronoid systems such as the Wiener index, variants of Szeged, Schultz and Gutman indices.

Quotient graphs for power graphs

In a previous paper of the first author a procedure was developed for counting the components of a graph through the knowledge of the components of one of its quotient graphs. Here we apply that procedure to the proper power graph $\mathcal{P}\_0(G)$ of a finite group $G$, finding a formula for the number of its components which is particularly illuminative when $G\leq S\_n$ is a fusion controlled permutation group. We make use of the proper quotient power graph $\widetilde{\mathcal{P}}\_0(G)$, the proper order graph $\mathcal{O}\_0(G)$ and the proper type graph $\mathcal{T}\_0(G)$. All those graphs are quotient of $\mathcal{P}\_0(G)$. We emphasize the strong link between them determining number and typology of the components of the above graphs for $G=S\_n$. In particular, we prove that the power graph $\mathcal{P}(S\_n)$ is $2$-connected if and only if the type graph $\mathcal{T}(S\_n)$ is $2$-connected, if and only if the order graph $\mathcal{O}(S\_n)$ is $2$-connected, that is, if and only if either $n = 2$ or none of $n, n-1$ is a prime.

Graph homomorphisms and components of quotient graphs

We study how the number c(X) of components of a graph X can be expressed through the number and properties of the components of a quotient graph X/\sim . We partially rely on classic qualifications of graph homomorphisms such as locally constrained homomorphisms and on the concept of equitable partition and orbit partition. We introduce the new definitions of pseudo-covering homomorphism and of component equitable partition, exhibiting interesting inclusions among the various classes of considered homomorphisms. As a consequence, we find a procedure for computing c(X) when the projection on the quotient X/\sim is pseudo-covering. That procedure becomes particularly easy to handle when the partition corresponding to X/\sim is an orbit partition.

Labelled quotient graphs and topological features in crystal structures

Labelled quotient graphs and topological features in crystal structures

Translation surfaces and the curve graph in genus two

Let $S$ be a (topological) compact closed surface of genus two. We associate to each translation surface $(X,\omega) \in \mathcal{H}(2)\sqcup\mathcal{H}(1,1)$ a subgraph $\hat{\mathcal{C}}_{\rm cyl}$ of the curve graph of $S$. The vertices of this subgraph are free homotopy classes of curves which can be represented either by a simple closed geodesic, or by a concatenation of two parallel saddle connections (satisfying some additional properties) on $X$. The subgraph $\hat{\mathcal{C}}_{\rm cyl}$ is by definition $\mathrm{GL}^+(2,\mathbb{R})$-invariant. Hence, it may be seen as the image of the corresponding Teichm\"uller disk in the curve graph. We will show that $\hat{\mathcal{C}}_{\rm cyl}$ is always connected and has infinite diameter. The group ${\rm Aff}^+(X,\omega)$ of affine automorphisms of $(X,\omega)$ preserves naturally $\hat{\mathcal{C}}_{\rm cyl}$, we show that ${\rm Aff}^+(X,\omega)$ is precisely the stabilizer of $\hat{\mathcal{C}}_{\rm cyl}$ in ${\rm Mod}(S)$. We also prove that $\hat{\mathcal{C}}_{\rm cyl}$ is Gromov-hyperbolic if $(X,\omega)$ is completely periodic in the sense of Calta. It turns out that the quotient of $\hat{\mathcal{C}}_{\rm cyl}$ by ${\rm Aff}^+(X,\omega)$ is closely related to McMullen's prototypes in the case $(X,\omega)$ is a Veech surface in $\mathcal{H}(2)$. We finally show that this quotient graph has finitely many vertices if and only if $(X,\omega)$ is a Veech surface for $(X,\omega)$ in both strata $\mathcal{H}(2)$ and $\mathcal{H}(1,1)$.

Partition distance in graphs

If G is a graph and \(\mathcal{P}\) is a partition of V(G), then the partition distance of G is the sum of the distance between all pairs of vertices that lie in the same part of \(\mathcal{P}\). This concept generalizes several metric concepts and is dual to the concept of the colored distance due to Dankelmann, Goddard, and Slater. It is proved that the partition distance of a graph can be obtained from the Wiener index of weighted quotient graphs induced by the transitive closure of the Djokovic–Winkler relation as well as by any partition coarser than it. It is demonstrated that earlier results follow from the obtained theorems. Applying the main results, upper bounds on the partition distance of trees with prescribed order and radius are proved and corresponding extremal trees characterized.

Growth and cogrowth of normal subgroups of a free group

We give a sufficient condition for a sequence of normal subgroups of a free group to have the property that both, their growths tend to the upper bound and their cogrowths tend to the lower bound. The condition is represented by planarity of the quotient graphs of the tree.

The Szeged index and the Wiener index of partial cubes with applications to chemical graphs

In this paper, we study the Szeged index of partial cubes and hence generalize the result proved by Chepoi and Klavžar, who calculated this index for benzenoid systems. It is proved that the problem of calculating the Szeged index of a partial cube can be reduced to the problem of calculating the Szeged indices of weighted quotient graphs with respect to a partition coarser than Θ-partition. Similar result for the Wiener index was recently proved by Klavžar and Nadjafi-Arani. Furthermore, we show that such quotient graphs of partial cubes are again partial cubes. Since the results can be used to efficiently calculate the Wiener index and the Szeged index for specific families of chemical graphs, we consider C4C8 systems and show that the two indices of these graphs can be computed in linear time.