Quotient Graph Research Articles

Computation of Wiener Descriptor for Melamine Cyanuric Acid Structure

This paper shows the technique for calculating the Wiener index of melamine cyanuric acid structure by dividing it into smaller vertex-weighted quotient graphs. Melamine cyanuric acid is a crystalline complex made up of a 1 : 1 combination of melamine and cyanuric acid. One of the topological indices that have been investigated the most, both theoretically and in terms of practical applications, is the Wiener index. It has a strong relationship with boiling points and other quantities of molecules. The computational techniques decrease the computational cost by selecting such as the range of values for the Wiener index, the variety and the size of atoms and bonds, and the number of cycles. In this work, we have computed the Wiener index for melamine cyanuric acid structures and also have done a correlation between numerical values and chemical quantities.

Improved multi-agent controllability processing technique based on equitable partition

In this paper, we study the controllability of multi-agent systems by equitable partition and automorphism. For the case that cells are incompletely connected outside but completely connected inside, a necessary condition for controllability is given from the perspective of the rank of connection matrix. For the case of multiple cells being completely connected outside and incompletely connected inside, in terms of the eigenvalues and eigenvectors of L and Lπ, several sufficient and necessary conditions for controllability are presented. Once the quotient graph is controllable under single input or all nodes in nontrivial cells are leaders, the lower bound of controllable subspace is determined. Finally, we give the gap between the necessary condition and the sufficient condition for controllability from the aspect of equitable partition. One highlight of the results in this paper is that we show sufficient conditions to judge controllability by equitable partition and automorphism, which, for specific cases, provides one method that how to break through the defect that equitable partition can only obtain necessary conditions.

On Molecular Structural Characterization of Cyclen Cored Dendrimers

Macromolecules are gaining much attention in various fields today. Dendrimers are artificially synthesized macromolecules by convergent or divergent approach. They are compact regular structures with spherical dimension and has a vast number of application in disparate fields such as drug delivery, material science, and biology, magnetic resonance imaging, an organic light-emitting device, etc. Determining the pharmacological, chemical, and biological characteristics of a substance necessitates a significant amount of effort. From the chemical graph of the dendrimer structure, we can infer those characteristics with the help of numerical descriptors known as the topological index. The Wiener and Szeged indices are two important distance-based topological indices applicable in nanoscience. The degree-based topological indices also have great importance and huge applications in structural chemistry. These indices together with graph entropy are found to be more effective and have found application in different sciences. In this work, the Wiener index, Szeged indices, Mostar indices, and Padmakar Ivan index for cyclen cored dendrimers are evaluated by converting the original graph into quotient graphs using Θ * - classes. This technique is applied since the regular cut method is a lengthy process while moving on to higher generations and also due to the presence of odd cycles in the structure. The degree-based indices and the degree-based graph entropies for the cyclen cored dendrimers are further studied. The comparison graphs with respect to the topological indices as well as graph entropies are also presented.

Distance Based Structural Descriptors of Non-conjugated Ethylene Oxide Dendritic Core Decorated with Tetraphenylethylene

Topological descriptors have drawn more interest recently because of the simplicity of generation and quick evaluation times, by avoiding time-consuming laboratory experiments. The distance based descriptors stand out as being pivotal in the QSAR and QSPR analysis of physico-chemical aspects. Dendrimers are branching polymeric molecules distinguished by its systematic growth pattern. Non-conjugated Ethylene oxide dendritic core attached with tetraphenylethylene is one such hyper branched polymer with strong solubility in a majority of organic solvents making them an active constituent for a variety of experimental purposes. No significant amount of research has been carried over based on the topological aspects of this dendrimer. This study effectively establishes the correlation between the chemical and physical attributes of the structure which can further aid in experimental research. In this work, various fundamental distance based descriptors of Ethylene oxide cored dendrimer have been evaluated by converting the original graph to quotient graphs using Djokovic̀-Winkler relation. Moreover, a graphical comparison of these descriptors has also been plotted to help comprehend the growth trend of these numerical values.

On genera containing non-split Eichler orders over function fields

Grothendieck-Birkhoff Theorem states that every finite dimensional vector bundle over the projective line P1 splits as the sum of one dimensional vector This can be rephrased, in terms of orders, as stating that all maximal orders over the projective line in a matrix algebra split. In this work we study the extent to which this result can be generalized to Eichler orders when the base field F is finite. To be precise, we characterize both the genera of Eichler orders containing only split orders and the genera containing only a finite number of non-split conjugacy classes. The latter characterization is given for arbitrary projective curves over F. The method developed here also allows us to compute quotient graphs for some subgroups of $PGL_2(F[t])$ of arithmetical interest. This paper includes material from the unpublished work Simultaneous diagonalization of vector bundles.

A new infinite family of star normal quotient graphs of twisted wreath type

We construct the first infinite families of locally 2-arc transitive graphs with the property that the automorphism group has two orbits on vertices and is quasiprimitive on exactly one orbit, of twisted wreath type. This work contributes to Giudici, Li and Praeger’s program for the classification of locally 2-arc transitive graphs by showing that the star normal quotient twisted wreath category also contains infinitely many graphs.

The Reibergram for immunoglobulin A in dogs: Evaluation of intrathecal IgA synthesis using a quotient graph in dogs with neurological diseases.

Increased cerebrospinal fluid (CSF) protein concentration is a common finding in neurological diseases of dogs. Distinguishing between intrathecally-produced proteins and proteins that have passed the blood-CSF barrier because of barrier disruption facilitates diagnosis. Albumin is a microprotein mainly produced extrathecally that can be used as a reference marker for blood-CSF barrier dysfunction. Develop a quotient graph based on the CSF/serum quotient of albumin and immunoglobulin A (IgA; Reibergram) to visualize intrathecal IgA synthesis and blood-CSF barrier dysfunction. Retrospective single-center cohort study. A hyperbolic function was developed using data from 6 healthy Beagles and 38 dogs with neurological diseases in which an isolated blood-CSF barrier dysfunction was expected. The function was validated using data from 10 dogs with expected intrathecal IgA synthesis and was visualized as a quotient graph. Finally, the graph was used to evaluate data of 118 dogs with various neurological diseases. Within the Reibergram, the function describes the upper values of physiological IgA quotients. It detects diseases with expected intrathecal IgA synthesis with higher sensitivity (85%) and specificity (89%) than the IgA index. The upper value of the physiological albumin quotient is 2.22 and detects diseases with expected blood-CSF barrier dysfunction (sensitivity: 81%; specificity: 88%). The canine Reibergram can detect blood-CSF barrier dysfunction and intrathecal IgA synthesis in the majority of cases. The graphical visualization simplifies data evaluation and makes it a feasible tool in routine CSF diagnostic testing.

Trace Formulas for Schrödinger Operators on a Lattice

We consider Schr\"odinger operators with periodic potentials on periodic discrete graphs. Their spectrum consists of a finite number of bands. We determine trace formulas for the Schr\"odinger operators. The proof is based on the decomposition of the Schr\"odinger operators into a direct integral and a specific representation of fiber operators. The traces of the fiber operators are expressed as finite Fourier series of the quasimomentum. The coefficients of the Fourier series are given in terms of the potentials and cycles in the quotient graph from some specific cycle sets. We also present the trace formulas for the heat kernel and the resolvent of the Schr\"odinger operators and the determinant formulas.

Evaluating Mineral Lattices as Evolutionary Proxies for Metalloprotein Evolution.

Protein coordinated iron-sulfur clusters drive electron flow within metabolic pathways for organisms throughout the tree of life. It is not known how iron-sulfur clusters were first incorporated into proteins. Structural analogies to iron-sulfide minerals present on early Earth, suggest a connection in the evolution of both proteins and minerals. The availability of large protein and mineral crystallographic structure data sets, provides an opportunity to explore co-evolution of proteins and minerals on a large-scale using informatics approaches. However, quantitative comparisons are confounded by the infinite, repeating nature of the mineral lattice, in contrast to metal clusters in proteins, which are finite in size. We address this problem using the Niggli reduction to transform a mineral lattice to a finite, unique structure that when translated reproduces the crystal lattice. Protein and reduced mineral structures were represented as quotient graphs with the edges and nodes corresponding to bonds and atoms, respectively. We developed a graph theory-based method to calculate the maximum common connected edge subgraph (MCCES) between mineral and protein quotient graphs. MCCES can accommodate differences in structural volumes and easily allows additional chemical criteria to be considered when calculating similarity. To account for graph size differences, we use the Tversky similarity index. Using consistent criteria, we found little similarity between putative ancient iron-sulfur protein clusters and iron-sulfur mineral lattices, suggesting these metal sites are not as evolutionarily connected as once thought. We discuss possible evolutionary implications of these findings in addition to suggesting an alternative proxy, mineral surfaces, for better understanding the coevolution of the geosphere and biosphere.

A graph-based approach for simultaneous semantic and instance segmentation of plant 3D point clouds.

Accurate simultaneous semantic and instance segmentation of a plant 3D point cloud is critical for automatic plant phenotyping. Classically, each organ of the plant is detected based on the local geometry of the point cloud, but the consistency of the global structure of the plant is rarely assessed. We propose a two-level, graph-based approach for the automatic, fast and accurate segmentation of a plant into each of its organs with structural guarantees. We compute local geometric and spectral features on a neighbourhood graph of the points to distinguish between linear organs (main stem, branches, petioles) and two-dimensional ones (leaf blades) and even 3-dimensional ones (apices). Then a quotient graph connecting each detected macroscopic organ to its neighbors is used both to refine the labelling of the organs and to check the overall consistency of the segmentation. A refinement loop allows to correct segmentation defects. The method is assessed on both synthetic and real 3D point-cloud data sets of Chenopodium album (wild spinach) and Solanum lycopersicum (tomato plant).

Group Consensus of Linear Multiagent Systems Under Nonnegative Directed Graphs

Group consensus implies reaching multiple groups where agents belonging to the same cluster reach state consensus. This article focuses on linear multiagent systems under nonnegative directed graphs. A new necessary and sufficient condition for ensuring group consensus is derived, which requires the spanning forest of the underlying directed graph and that of its quotient graph induced with respect to a clustering partition to contain equal minimum number of directed trees. This condition is further shown to be equivalent to containing cluster spanning trees, a commonly used topology for the underlying graph in the literature. Under a designed controller gain, lower bound of the overall coupling strength for achieving group consensus is specified. Moreover, the pattern of the multiple consensus states formed by all clusters is characterized when the overall coupling strength is large enough.

Quotients of the Bruhat-Tits tree by arithmetic subgroups of special unitary groups

Let $K$ be the function field of a curve $C$ over a field $\mathbb{F}$ of either odd or zero characteristic. Following the work by Serre and Mason on $\mathrm{SL}_2$, we study the action of arithmetic subgroups of $\mathrm{SU}(3)$ on its corresponding Bruhat-Tits tree associated to a suitable completion of $K$. More precisely, we prove that the quotient graph "looks like a spider", in the sense that it is the union of a set of cuspidal rays (the "legs"), parametrized by an explicit Picard group, that are attached to a connected graph (the "body"). We use this description in order to describe these arithmetic subgroups as amalgamated products and study their homology. In the case where $\mathbb{F}$ is a finite field, we use a result by Bux, K\"ohl and Witzel in order to prove that the "body" is a finite graph, which allows us to get even more precise applications.

On stabilizability of multi-agent systems

As an efficient collective response of multi-agent systems, the realization of stabilizability needs appropriate information transfer among agents, thus the topology structure is paramount. To investigate the influences of network topology on stabilizability, the concept of quotient graph (equivalent transformation of original graph) is proposed to realize the stabilizability of large-scale network systems which can be reduced to a small-scale quotient graph. As a cornerstone of topology graph transformation, NSi graph division, which is proposed for the first time in this paper, plays a crucial role in stabilizability analysis and has far-reaching impact. It establishes an equivalent connection between the stabilizability of the original topology graph and the quotient graph. Results reveal a nontrivial relationship between stabilizability of uncontrollable multi-agent systems and network topology (i.e., the stabilizability of uncontrollable multi-agent systems is not only related to the selection of leaders, but also depends on uncontrollable modes.), which is established by quotient graph as a visual generalization of the original network topology. Surprisingly, structural balance is not always good, and unbalance has a positive effect on the stabilizability of the system. Finally, the simulation experiments demonstrate the effectiveness of the theoretical results.

Combinatorial aspects of the Löwenstein avoidance rule. Part II: local and global constraints.

The determination of the independence ratio of a periodic net requires finding a subgroup of the translation group of the net for which the quotient graph and a fundamental transversal have the same independence ratio; the respective motif defines a periodic factor of the net. This article deals with practical issues regarding the calculation of the independence ratio of mainly 2-periodic nets, with an application to the 200 2-periodic nets listed on the RCSR (Reticular Chemistry Structure Resource) site. A companion paper described a calculation technique of independence ratios of finite graphs based on propositional calculus. This paper focuses on criteria for the choice of the translation subgroup and of the transversal. The translation subgroup should be chosen in such a way as to eliminate every cycle in the quotient graph that is shorter than structural cycles, or rings, of the net. Topological constraints provide an upper bound to the independence ratio of 2-periodic nets and mostly enable the determination of the associated factor, thus giving a description of a periodic distribution in saturated solid solutions obeying some avoidance rule.

Structure and behavior in Boolean monotonic model pools

One of the main theoretical questions in the field of discrete regulatory networks is the question what aspects of the dynamics – the structure of the state transition graph – are already imposed by structural descriptions of the network such as the interaction graph. For Boolean networks, prior work has concentrated on different versions of the Thomas conjectures that link feedback cycles in the network structure to attractor properties. Other approaches check algorithmically whether certain properties hold true for all models sharing specific structural constraints, e.g. by using model checking techniques. In this work we investigate the behavior of the pool of Boolean networks in agreement with a given interaction graph using a different approach. Grouping together states that are updated consistently across the pool we derive an equivalence relation and analyze a corresponding quotient graph on the state space. By construction this graph yields information about the dynamics of all functions in the pool. Our main result is that this graph can be computed efficiently without enumerating and analyzing all individual functions. This opens up new possibilities for applications, where such model pools arise when modeling under uncertainty.

Knot graphs and Gromov hyperbolicity

We define a broad class of graphs that generalize the Gordian graph of knots. These knot graphs take into account unknotting operations, the concordance relation, and equivalence relations generated by knot invariants. We prove that overwhelmingly, the knot graphs are not Gromov hyperbolic, with the exception of a particular family of quotient knot graphs. We also investigate the property of homogeneity, and prove that the concordance knot graph is homogeneous. Finally, we prove that that for any n, there exists a knot K such that the ball of radius n in the Gordian graph centered at K contains no connected sum of torus knots.

Two-sided estimates of total bandwidth for Schrödinger operators on periodic graphs

<p style='text-indent:20px;'>We consider Schrödinger operators with periodic potentials on periodic discrete graphs. Their spectrum consists of a finite number of bands. We obtain two-sided estimates of the total bandwidth for the Schrödinger operators in terms of geometric parameters of the graph and the potentials. In particular, we show that these estimates are sharp. It means that these estimates become identities for specific graphs and potentials. The proof is based on the Floquet theory and trace formulas for fiber operators. The traces are expressed as finite Fourier series of the quasimomentum with coefficients depending on the potentials and cycles of the quotient graph from some specific cycle sets. In order to obtain our results we estimate these Fourier coefficients in terms of geometric parameters of the graph and the potentials.</p>

Edge-Transitive Cyclic Covers of Complete Graphs with Prime Power Order

Characterizing regular covers of symmetric graphs is one of the fundamental topics in the field of algebraic graph theory, and is often a key step for approaching general symmetric graphs. Complete graphs, which are typical symmetric graphs, naturally appear in the study of many symmetric graphs as normal quotient graphs. In this paper, a characterization of edge-transitive cyclic covers of complete graphs with prime power order is given by using the techniques of finite group theory and the related properties of coset graphs. Certain previous results are generalized and some new families of examples are founded.

Maximal orders containing a given sub-order for local fields of even characteristic

We extend, to the case of local fields of even characteristic, our previous computations for the set of maximal orders containing given quaternions as concrete sub-graphs of the Bruhat–Tits tree. Computing the relative position of such sub-graphs, the branches, is useful, for instance, to explicitly compute relative spinor images, thus solving the selectivity problem. They also play a role in the description of quotient graphs, which are useful to study matrix groups of arithmetical importance. As in earlier work, we concentrate on orders generated by two quaternions, but extending our method to larger generating sets is quite straightforward. In our previous work, the results were given mainly in terms of the quadratic defect. In the present context, we introduce and characterize an analogous concept for Artin–Scheier extensions, to take care of quaternions generating Galois field extensions. The latter extensions have no non-trivial elements of null trace. For this reason, we state our results in terms of a wider family of algebra presentations, in contrast to the aforementioned work, where we chose a pair of pure quaternions as generators of the order.

Trace formulas for Schrödinger operators on periodic graphs

We consider Schrödinger operators with periodic potentials on periodic discrete graphs. Their spectrum consists of a finite number of bands. We determine trace formulas for the Schrödinger operators. The proof is based on the decomposition of the Schrödinger operators into a direct integral and a specific representation of fiber operators. The traces of the fiber operators are expressed as finite Fourier series of the quasimomentum. The coefficients of the Fourier series are given in terms of the potentials and cycles in the quotient graph from some specific cycle sets. We also present the trace formulas for the heat kernel and the resolvent of the Schrödinger operators and the determinant formulas.