Quotient Graph Research Articles

Topological characterization, entropy measures and prediction of properties of Iridium cored dendrimer

This study presents a comprehensive structural analysis of Iridium-cored dendrimers, denoted by Inmlr;n≥1, with an emphasis on degree and distance based metrics to elucidate their structure-property relationships. Dendrimers, known for their highly branched architectures, are versatile macromolecules with applications across various scientific fields. In this work, molecular descriptors serve as essential numerical indicators, capturing bonding characteristics and aiding in the prediction of material properties. Leveraging a novel quotient graph approach, we compute a range of distance based indices, including the Wiener (W), Szeged (Sz and Sze), Mostar (Mo and Moe), and Padmakar-Ivan (PI) indices. We further derive generalized expressions for entropy measures associated with degree based indices, such as Shannon's entropy, providing a robust framework for assessing the structural complexity of these compounds. Additionally, we examine various degree based descriptors with entropy measures—including the Zagreb (M1, M2, and HM), Harmonic (H), Forgotten (F), Randic (R and RR), ABC, GA, SC, σ, and Irregularity indices (irr). A linear regression analysis is conducted to model dendrimer properties and forecast attributes for subsequent generations, potentially minimizing the need for extensive laboratory experimentation. Our findings provide valuable insights into the molecular intricacies of Iridium-cored dendrimers, bridging theoretical chemistry with practical applications and contributing to future advancements in materials science and engineering.

On the automorphisms of the Drinfeld modular groups

Let [Formula: see text] be the ring of elements in an algebraic function field [Formula: see text] over [Formula: see text] which are integral outside a fixed place [Formula: see text]. In contrast to the classical modular group [Formula: see text] and the Bianchi groups, the Drinfeld modular group [Formula: see text] is not finitely generated and its automorphism group [Formula: see text] is uncountable. Except for the simplest case [Formula: see text] not much is known about the generators of [Formula: see text] or even its structure. We find a set of generators of [Formula: see text] for a new case. On the way, we show that every automorphism of [Formula: see text] acts on both the cusps and the elliptic points of [Formula: see text]. Generalizing a result of Reiner for [Formula: see text] we describe for each cusp an uncountable subgroup of [Formula: see text] whose action on [Formula: see text] is essentially defined on the stabilizer of that cusp. In the case where [Formula: see text] (the degree of [Formula: see text]) is [Formula: see text], the elliptic points are related to the isolated vertices of the quotient graph [Formula: see text] of the Bruhat–Tits tree. We construct an infinite group of automorphisms of [Formula: see text] which fully permutes the isolated vertices with cyclic stabilizer.

Contagion probability in linear threshold model

We study a linear threshold model on a simple undirected connected network G where each non-seed becomes active if and only if the proportion of its active neighbors exceeds its adoption threshold. Each threshold function ϕ:V→[0,1] is viewed as a point (ϕ(v1),…,ϕ(vn)) in the n-cube [0,1]n, where V={v1,…,vn} is the set of nodes in G. We define ϕ as a contagious point of a subset S of nodes if it can induce full contagion from S. Consequently, the volume of the set of contagious points of S in [0,1]n represents the probability of full contagion from S when the adoption threshold of each node is independently and uniformly distributed in [0,1], which we term the contagion probability of S and denote by pc(S). We derive an explicit formula for pc(S), showing that pc(S) is determined by how likely S can produce full contagion exclusively through each spanning tree of the quotient graph GS of G in which S is treated as a single node. Besides, we compare pc(S) with the contagion threshold of S, which is denoted by qc(S) and is the probability of full contagion from S when all nodes share a common adoption threshold q chosen uniformly at random from [0,1]. We show that the presence of a cycle in GS is necessary but not sufficient for pc(S) to exceed qc(S), which indicates that allowing threshold heterogeneity may not always increase the chance of full contagion. Our framework can be extended to study contagion under various threshold settings.

Molecular descriptors of symmetrically configured carbon nanocones via quotient graph technique

Carbon nanocones, characterized by planar networks primarily consisting of hexagonal carbon faces and a few pentagons in the core, hold significant promise in various fields such as electronics, medicine, and material research due to their exceptional properties. This study exclusively investigates the topological descriptors of symmetrically configured carbon nanocones, specifically nanocones with two and three pentagonal core respectively. By modeling these structures as chemical graphs and employing molecular descriptors, we aim to provide a quantitative analysis of the structures without laboratory experimentation. The quotient graph approach is employed to determine the distance based descriptors, namely wiener, szeged and Padmakar-Ivan indices for these symmetrically configured carbon nanocones. The analytically closed formulas determined could be useful in quantitative structure-property-activity relationship studies, which helps in predicting the physico-chemical properties of the underlying structures. Thus the findings in this work not only deepen our understanding of symmetric carbon nanocones but also pave the way for significant advancements in nanotechnology and materials science.

An Alternate Form of Merino-Mǐcka-Mütze’s Solution to a Knuth’s Combinatorial Generation Problem

A modification of Merino-Mǐcka-Mütze’s solution to a combinatorial generation problem of Knuth is proposed in this survey. The resulting alternate form to such solution is compatible with a reinterpretation by the author of a proof of existence of Hamilton cycles in the middle-levels graphs. Such reinterpretation is given in terms of a dihedral quotient graph associated to each middle-levels graph. The vertices of such quotient graph represent Dyck words and their associated ordered trees. Those Dyck words are linearly ordered via a rooted tree that covers all their tight, or irreducible, forms, offering an universal reference point of view to express and integrate the periodic paths, or blocks, whose concatenation leads to Hamilton cycles resulting from the said solution.

Molecular Descriptors Of Certain Class Of Carbon Nanocone Networks Through Quotient Graph Approach

Nanoparticles have potential applications in a wide range of fields, including electronics, medicine and material research, because of their remarkable and exceptional attributes. Carbon nanocones are planar carbon networks with mostly hexagonal faces and a few non-hexagonal faces (mostly pentagons) in the core. Two types of nanocone configurations are possible: symmetric and asymmetric, depending on where the pentagons are positioned within the structure. In addition to being a good substitute for carbon nanotubes, carbon nanocones have made an identity for themselves in a number of fields, including biosensing, electrochemical sensing, biofuel cells, supercapacitors, gas storage devices, and biomedical applications. Their astonishing chemical and physical attributes have made them well-known and widely accepted in the fields of condensed matter physics, chemistry, material science, and nanotechnology. Mathematical and chemical breakthroughs were made possible by the concept of modeling a chemical structure as a chemical graph and quantitatively analyzing the related graph using molecular descriptors. Molecular descriptors are useful in many areas of chemistry, biology, computer science, and other sciences because they allow for the analysis of chemical structures without the need for experiments. In this work, the quotient graph approach is used to establish the distance based descriptors of symmetrically configured two-pentagonal and three-pentagonal carbon nanocones.

Multi-type synchronization for coupled van der Pol oscillator systems with multiple coupling modes.

In this paper, we investigate synchronous solutions of coupled van der Pol oscillator systems with multiple coupling modes using the theory of rotating periodic solutions. Multiple coupling modes refer to two or three types of coupling modes in van der Pol oscillator networks, namely, position, velocity, and acceleration. Rotating periodic solutions can represent various types of synchronous solutions corresponding to different phase differences of coupled oscillators. When matrices representing the topology of different coupling modes have symmetry, the overall symmetry of the oscillator system depends on the intersection of the symmetries of the different topologies, determining the type of synchronous solutions for the coupled oscillator network. When matrices representing the topology of different coupling modes lack symmetry, if the adjacency matrices representing different coupling modes can be simplified into structurally identical quotient graphs (where weights can be proportional) through the same external equitable partition, the symmetry of the quotient graph determines the synchronization type of the original system. All these results are consistent with multi-layer networks where connections between different layers are one-to-one.

Quotient graphs of symmetrically rigid frameworks

A natural problem in combinatorial rigidity theory concerns the determination of the rigidity or flexibility of bar-joint frameworks in \mathbb{R}^{d} that admit some non-trivial symmetry. When d=2 there is a large literature on this topic. In particular, it is typical to quotient the symmetric graph by the group and analyse the rigidity of symmetric, but otherwise generic frameworks, using the combinatorial structure of the appropriate group-labelled quotient graph. However, mirroring the situation for generic rigidity, little is known combinatorially when d\geq 3 . Nevertheless in the periodic case, a key result of Borcea and Streinu in 2011 characterises when a quotient graph can be lifted to a rigid periodic framework in \mathbb{R}^{d} . We develop an analogous theory for symmetric frameworks in \mathbb{R}^{d} . The results obtained apply to all finite and infinite 2-dimensional point groups, and then in arbitrary dimension they concern a wide range of infinite point groups, sufficiently large finite groups and groups containing translations and rotations. For the case of finite groups we also derive results concerning the probability of assigning group labels to a quotient graph so that the resulting lift is symmetrically rigid in \mathbb{R}^{d} .

Computation of Structural Descriptors of Pyrene Cored Dendrimers through Quotient Graph Approach and Its Graph Entropy Measures

Dendrimers are highly defined hyperbranched artificial macromolecules, synthesised by convergent or divergent approach with specific applications in various fields. Dendrimers can be represented as graph models, from which a quantitative description can be drawn in relation with their structural properties. The distance-based and the degree-based descriptors have great importance and huge applications in structural chemistry. These indices together with entropy measures are found to be more effective and have found application in scientific fields. The idea of graph entropy is to characterise the complexity of graphs. The use of these graph invariants in quantitative structure property relationship and quantitative structure activity relationship studies has become of major interest in recent years. In this paper, the distance-based molecular descriptors of pyrene cored dendrimers are studied applying the technique of converting original graph into quotient graphs using Θ∗-classes. It is to be noted that, since the pyrene cored dendrimer, Gn is not a partial cube, usual cut method is not applicable. Further, various degree-based descriptors and their corresponding graph entropies of the pyrene cored dendrimers are also studied. Based on the obtained results, a comparative analysis as well as a regression analysis was carried out.

Quasi-inner automorphisms of Drinfeld modular groups

Let A be the set of elements in an algebraic function field K over \mathbb{F}_{q} which are integral outside a fixed place \infty . Let G=\mathrm{GL}_{2}(A) be a Drinfeld modular group . The normalizer of G in \mathrm{GL}_{2}(K) , where K is the quotient field of A , gives rise to automorphisms of G , which we refer to as quasi-inner . Modulo the inner automorphisms of G , they form a group \mathrm{Quinn}(G) which is isomorphic to \mathrm{Cl}(A)_{2} , the 2 -torsion in the ideal class group \mathrm{Cl}(A) . The group \mathrm{Quinn}(G) acts on all kinds of objects associated with G . For example, it acts freely on the cusps and elliptic points of G . If \mathcal{T} is the associated Bruhat–Tits tree, the elements of \mathrm{Quinn}(G) induce non-trivial automorphisms of the quotient graph G \backslash \mathcal{T} , generalizing an earlier result of Serre. It is known that the ends of G \backslash \mathcal{T} are in one-to-one correspondence with the cusps of G . Consequently, \mathrm{Quinn}(G) acts freely on the ends. In addition, \mathrm{Quinn}(G) acts transitively on those ends which are in one-to-one correspondence with the vertices of G \backslash \mathcal{T} whose stabilizers are isomorphic to \mathrm{GL}_{2}(\mathbb{F}_{q}) .

Maximal independence and symmetry in crystal chemistry of natural tectosilicates.

Löwenstein's avoidance rule in aluminosilicates is reinterpreted on the basis of the fourth Pauling rule. It is shown that avoidance of Si-O-Si bridges may account for avoidance of Al-O-Al bridges. In view of this interpretation, it is proposed that the most favourable distributions of cations entering in substitution of silicon in the framework are associated to maximal independent sets of the respective 3-periodic nets. Among all possible solutions, only those with maximal symmetry are realized. The applicability of the concept is demonstrated for a few natural tectosilicates, which have been analysed through the prism of their labelled quotient graph.

Cut method for the multivariable Szeged‐like polynomial with application to hyaluronic acid

AbstractSzeged‐like topological indices are commonly studied distance‐based molecular descriptors. These indices, together with related graph polynomials, offer valuable tools for characterising graph structures. Two new polynomials, the SMP polynomial and the edge‐SMP polynomial, have been recently introduced and allow efficient computation of Szeged, Mostar, and PI indices (note that the abbreviation SMP comes from the names of these indices). In this paper, we develop the cut method to compute different versions of the SMP polynomial in a much more general way. In particular, the multivariable Szeged‐like polynomial is introduced for any strength‐weighted graph. Moreover, it is shown how it can be computed using strength‐weighted quotient graphs obtained by a partition of the edge set coarser than the ‐partition. The developed approach is demonstrated by calculating weighted‐plus SMP polynomial and related topological indices for hyaluronic acid.

Graphic statics and symmetry

Reciprocal diagrams are a geometric construction dating back to Maxwell and Cremona in which a self-stressed plane framework with a planar graph is paired with another self-stressed reciprocal framework on the dual graph. Either one of the reciprocal frameworks is the form diagram of a self-stressable structure and the other is the force diagram of the corresponding axial forces. This geometric technique offers insights into the self-stresses and infinitesimal motions (mechanisms) of both frameworks in the reciprocal pair. For a symmetric framework with a fully-symmetric self-stress, we obtain an equi-symmetric reciprocal pair of plane frameworks, as well as the associated symmetric discrete dual Airy stress function polyhedra. In this paper we exploit symmetry to refine the Maxwell–Cremona correspondence by considering the decomposition of the self-stress and motion spaces into invariant subspaces corresponding to the irreducible representations of the symmetry group. As such, the familiar s=m∗+1 relationship for the number of self-stresses of a framework, s, and the number of mechanisms of the reciprocal, m∗, is reworked into a symmetry adapted version which provides greater insights into the properties of the reciprocal framework pair. We also show how the quotient graph of a symmetric framework and its reciprocal can be used to efficiently detect infinitesimal motions, self-stresses and polyhedral liftings of different symmetry types. This allows for symmetry-adapted simplified structural analyses of symmetric structures.

Quotient of quotient graph a novel approach to compute π‐conjugated dendrimer and predict its properties

AbstractThe role of dendrimers is getting updated and grabbing immense attention in many fields. Dendrimers are used in a broad spectrum, in specific, it is widely utilized in drug delivery, for targeted carrier and specific action. Also, in drug catalysis, it improves the solubility of poorly soluble drugs and increases the stability of active ingredients within the cores. Many researchers are interested in developing new dendrimers by using various compounds. ‐conjugated dendrimers via truxenes and thienylethynylene units, are considered a promising material in the field of medicinal chemistry, bio‐organic, and environmental sciences. In QSAR/QSPR, the structural analysis of chemical compounds is examined by topological descriptors. In this article, the various distance‐based topological indices such as Wiener, Szeged, and Mostar of gradient star‐persistent ‐conjugated dendrimers are determined. Since gradient star‐persistent ‐conjugated dendrimers have odd cycles, the usual cut method is not applicable, therefore we use a new concept, the quotient of quotient graph to determine the numerical expression of various indices. An analysis of linear regression reveals that degree‐based topological descriptors predict better physicochemical properties.

Trace formulas for magnetic Schrödinger operators on periodic graphs and their applications

We consider Schrödinger operators with periodic magnetic and electric potentials on periodic discrete graphs. The spectrum of such operators consists of a finite number of bands. Some of them and even all may be degenerate. We determine trace formulas for the magnetic Schrödinger 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 magnetic fluxes, electric potentials and cycles in the quotient graph from some specific cycle sets. Using the trace formulas we obtain new lower estimates of the total bandwidth for the magnetic Schrödinger operator in terms of geometric parameters of the graph, magnetic fluxes and electric potentials. We show that these estimates are sharp.

How Many Cliques Can a Clique Cover Cover?

This work examines the problem of clique enumeration on a graph by exploiting its clique covers. The principle of inclusion/exclusion is applied to determine the number of cliques of size $r$ in the graph union of a set $\mathcal{C} = \{c_1, \ldots, c_m\}$ of $m$ cliques. This leads to a deeper examination of the sets involved and to an orbit partition, $\Gamma$, of the power set $\mathcal{P}(\mathcal{N}_{m})$ of $\mathcal{N}_{m} = \{1, \ldots, m\}$. Applied to the cliques, this partition gives insight into clique enumeration and yields new results on cliques within a clique cover, including expressions for the number of cliques of size $r$ as well as generating functions for the cliques on these graphs. The quotient graph modulo this partition provides a succinct representation to determine cliques and maximal cliques in the graph union. The partition also provides a natural and powerful framework for related problems, such as the enumeration of induced connected components, by drawing upon a connection to extremal set theory through intersecting sets.

Computing Molecular Descriptors of Polyphenylene Dendrimer

Dendrimers are hyperbranched monodisperse macromolecules with constraint size and are shape persistent. Topological indices are numerical parameters that provide a correlation with the chemical, physical, and biological properties of the molecular structure. We have computed distance based molecular descriptors of polyphenylene dendrimer using the technique, quotient of quotient graphs, and analyzed a graphical comparison. This technique helps in obtaining numerical expressions of molecular descriptors on complex chemical structures. Our results help in better understanding of the properties of polyphenylene dendrimer which can be used in drug design.

FunQG: Molecular Representation Learning via Quotient Graphs.

To accurately predict molecular properties, it is important to learn expressive molecular representations. Graph neural networks (GNNs) have made significant advances in this area, but they often face limitations like neighbors-explosion, under-reaching, oversmoothing, and oversquashing. Additionally, GNNs tend to have high computational costs due to their large number of parameters. These limitations emerge or increase when dealing with larger graphs or deeper GNN models. One potential solution is to simplify the molecular graph into a smaller, richer, and more informative one that is easier to train GNNs. Our proposed molecular graph coarsening framework called FunQG, uses Functional groups as building blocks to determine a molecule's properties, based on a graph-theoretic concept called Quotient Graph. We show through experiments that the resulting informative graphs are much smaller than the original molecular graphs and are thus more suitable for training GNNs. We apply FunQG to popular molecular property prediction benchmarks and compare the performance of popular baseline GNNs on the resulting data sets to that of state-of-the-art baselines on the original data sets. Our experiments demonstrate that FunQG yields notable results on various data sets while dramatically reducing the number of parameters and computational costs. By utilizing functional groups, we can achieve an interpretable framework that indicates their significant role in determining the properties of molecular quotient graphs. Consequently, FunQG is a straightforward, computationally efficient, and generalizable solution for addressing the molecular representation learning problem.

Termination of amnesiac flooding

We consider a stateless ‘amnesiac’ variant of the stateful distributed network flooding algorithm, expanding on our conference papers [PODC’19, STACS’20]. Flooding begins with a set of source ‘initial’ nodes I seeking to broadcast a message M in rounds, in a network represented by an undirected graph (G, E) with set of nodes G and edges E. In every round, nodes with M send M to all neighbours from which they did not receive M in the previous round. Nodes do not remember earlier rounds, raising the possibility that M circulates indefinitely. Stateful flooding overcomes this by nodes recording every message circulated and ignoring M if received again. This overhead was assumed to be necessary. We show that almost optimal broadcast can still be achieved without this overhead. We prove that amnesiac flooding terminates on every finite graph and derive sharp bounds for termination times. Define (G, E) to be I-bipartite if the quotient graph, contracting all nodes in I to a single node, is bipartite. We prove that, if d is the diameter and e(I) the eccentricity of the set I, flooding terminates in e(I) rounds if (G, E) is I-bipartite and j rounds with e(I) < jle e(I)+d+1 le 2d +1 if (G, E) is non I-bipartite. The separation in the termination times can be used for distributed discovery of topologies/distances in graphs. Termination is guaranteed if edges are lost during flooding but not, in general, if there is a delay at an edge. However, the cases of single-edge fixed delays of duration tau rounds in single-source bipartite graphs terminate by round 2d+tau -1, and all cases of multiple-edge fixed delays in multiple-source cycles terminate.

Generalized Cut Method for Computing Szeged–Like Polynomials with Applications to Polyphenyls and Carbon Nanocones

Szeged, Padmakar-Ivan (PI), and Mostar indices are some of the most investigated distance-based Szeged-like topological indices. On the other hand, the polynomials related to these topological indices were also introduced, for example the Szeged polynomial, the edgeSzeged polynomial, the PI polynomial, the Mostar polynomial, etc. In this paper, we introduce a concept of the general Szeged-like polynomial for a connected strength-weighted graph. It turns out that this concept includes all the above mentioned polynomials and also infinitely many other graph polynomials. As the main result of the paper, we prove a cut method which enables us to efficiently calculate a Szeged-like polynomial by using the corresponding polynomials of strength-weighted quotient graphs obtained by a partition of the edge set that is coarser than Θ∗ -partition. To the best of our knowledge, this represents the first implementation of the famous cut method to graph polynomials. Finally, we show how the deduced cut method can be applied to calculate some Szeged-like polynomials and corresponding topological indices of para-polyphenyl chains and carbon nanocones.