Abstract A graph is distance magic if it admits a bijective labeling of its vertices by integers from 1 up to the order of the graph in such a way that the sum of the labels of all the neighbors of a vertex is independent of a given vertex. We introduce the concept of a self-reverse distance magic labeling of a regular graph which allows for a more compact description of the graph and the labeling in terms of the corresponding quotient graph. We show that the members of several known infinite families of tetravalent distance magic graphs admit such labelings. We present a novel general construction producing a new distance magic graph from two existing ones. Using it we show that for each integer $$n \ge 6$$ n ≥ 6 , except for the odd integers up to 19, there exists a connected tetravalent graph of order n admitting a self-reverse distance magic labeling. We also determine all connected tetravalent graphs up to order 30 admitting a self-reverse distance magic labeling. The obtained data suggests a number of natural interesting questions giving several possibilities for future research.

Multipartite entangled states are essential for multiuser quantum cryptography. While large-scale continuous-variable (CV) cluster states, particularly the dual-rail cluster state, have been well studied in measurement-based quantum computation, their cryptographic potential remains underexplored. Here, we propose a three-user conference key protocol using a CV dual-rail cluster state. By applying a node-coloring scheme to the infinite dual-rail graph, we create a six-mode pure graph state ideal for cryptographic tasks. Our results demonstrate near-GHZ (Greenberger-Horne-Zeilinger) performance for quantum conference key agreement (QCKA). Crucially, our protocol uniquely enables bipartite keys post-QCKA, which GHZ states cannot provide. It also surpasses two-mode squeezed vacuum states in generating bipartite keys within downstream-access networks. Considering finite-size effects and impure squeezed states, our scheme remains robust despite experimental imperfections. We also introduce an enhanced method to more accurately estimate bipartite key generation capacity in quantum networks, paving the way for practical multiuser quantum cryptography.

Continuum robots (CR) can achieve excellent dexterity and flexibility, making them suitable for navigating through cluttered environments and safely interacting with obstacles. Due to the underactuated nature of CRs, the contact mode between the robot and environment affects the static robot configuration. We show that the configuration space topology induced by environmental obstacles can be characterized by a quotient structure with a quotient space consisting of zero-actuation configurations. We propose to use the quotient space as a road map for motion planning to reduce computational load for exploration. Specifically, we propose an algorithm that identifies the quotient space as a graph of configuration modes by constructing a graph of convex sets in the free workspace, conducting tree search and convex optimizations to find candidate configurations, and then using elastic energy minimization to find the modes. We then use a motion planner which finds a path in the quotient space graph and constructs a continuous path in the configuration space. We demonstrate our method in several complex 3D environments and show that our method outperforms baselines in terms of computation time and success rate.

Random generation of crystal structures is a key to the success of predicting unknown crystals. In this work, we introduce a general method for refining and selecting random structures that relies on minimal prior information. The method establishes a quotient graph from the random structure using a near-neighbor finding algorithm, which subsequently guides the refinement of the initial structure. To validate this approach, we apply it to nine distinct systems, and the outcomes indicate that it effectively yields a great number of low-energy structures. This technique could be integrated into most structure prediction algorithms to generate more sound initial structures, thereby expediting the search for ground state structures.

Almost equitable partitions (AEPs) have been linked to cluster synchronization in oscillatory systems, highlighting the importance of structure in collective network dynamics. We provide a general spectral framework that formalizes this connection, showing how eigenvectors associated with AEPs span a subspace of the Laplacian spectrum that governs partition-induced synchronization behavior. This offers a principled reduction of network dynamics, allowing clustered states to be understood in terms of quotient graph projections. Our approach clarifies the conditions under which transient hierarchical clustering and multi-frequency synchronization emerge and connects these dynamical phenomena directly to network symmetry and community structure. In doing so, we bridge a critical gap between static topology and dynamic behavior, namely, the lack of a spectral method for analyzing synchronization in networks that exhibit exact or approximate structural regularity. Perfect AEPs are rare in real-world networks since most have some degree of irregularity or noise. We define relaxation of an AEP we call a quasi-equitable partition at level δ (δ-QEP). δ-QEPs can preserve many of the clustering-relevant properties of AEPs while tolerating structural imperfections and noise. This extension enables us to describe synchronization behavior in more realistic scenarios, where ideal symmetries are rarely present. Our findings have important implications for understanding synchronization patterns in real-world networks, from neural circuits to power grids.

As an extension of a previous work, we analyse ordered and disordered Si/Al distributions in a few tectosilicates. The method is based on an analysis of the inter-relations between maximal independent sets in a labelled quotient graph of the net. The analysis suggests the presence of specific disordered substructures, called here alveoli, coexisting with fully ordered parts, defining partial order in submicroscopic domains. The method is first illustrated with bikitaite, chabazite and analcime and fully developed for natural zeolites with the GIS framework type. We show that the principle of maximal independence applied to the labelled quotient graph of the net gis can be used to justify composition and order/disorder of the two ordered phases gismondine and amicite and of the disordered phases garronite and gobbinsite. To this list of natural zeolites, we add the disordered, synthetic Na-P2 phase. The results are in good agreement with published 29Si magic-angle spinning NMR data for bikitaite, chabazite and analcime.

The unions of directed graphs are the simplest examples of pushouts of directed graphs. The conditions under which they contravariantly induce surjective gauge-equivariant pullbacks of graph \mathrm{C}^{*} -algebras have been well studied and vastly instantiated in noncommutative topology (e.g., quantum balls and spheres). Herein, we go beyond the unions of graphs to systematically determine optimal conditions for more general length-preserving pushouts of graphs under which they contravariantly induce graded pullbacks of path algebras, Leavitt path algebras, and graph \mathrm{C}^{*} -algebras. Our pullbacks are surjective only on one side, as dictated by natural examples and K-theory. The proposed new approach enlarges the scope of applications from admissible subgraphs (also called quotient graphs) to generalizations of unlabeled foldings of Stallings and collapsing the line graphs of graphs to initial graphs. Moreover, we introduce the concept of locally derived graphs, which substantially extends the paradigm of derived graphs (or skew products of graphs), and use the projection foldings from locally derived graphs to their base (or voltage) graphs to obtain one-surjective pullbacks of graph \mathrm{C}^{*} -algebras.

This paper introduces a framework of hypercompositional algebra on fuzzy graphs by defining and analyzing fuzzy path-based hyperoperations. Building on the notion of strongest strong paths (paths that are both strength-optimal and composed exclusively of strong edges, where each edge achieves maximum connection strength between its endpoints), we define two operations: a vertex-based fuzzy path hyperoperation and an edge-based variant. These operations generalize classical graph hyperoperations to the fuzzy setting while maintaining compatibility with the underlying topology. We prove that the vertex fuzzy path hyperoperation is associative, forming a fuzzy hypersemigroup, and establish additional properties such as reflexivity and monotonicity with respect to α-cuts. Structural features such as fuzzy strong cut vertices and edges are examined, and a fuzzy distance function is introduced to quantify directional connectivity strength. We define an equivalence relation based on mutual full-strength reachability and construct a quotient fuzzy graph that reflects maximal closed substructures under the vertex fuzzy path hyperoperation. Applications are discussed in domains such as trust networks, biological systems, and uncertainty-aware communications. This work aims to lay the algebraic foundations for further exploration of fuzzy hyperstructures that support modeling, analysis, and decision-making in systems governed by partial and asymmetric relationships.

We present an operator theoretic coarse-graining (or model order reduction) procedure for stochastic matrices by clustering. The method is consistent with the natural structure of Markov theory, preserving positivity and mass, and does not rely on any tools from Hilbert space theory. The reconstruction is provided by a generalized Penrose–Moore inverse of the coarse-graining operator incorporating the inhomogeneous invariant measure of the Markov matrix. As we will show, the method provides coarse-graining and reconstruction also on the level of tensor spaces, which is consistent with the notion of an incidence matrix and quotient graphs, and, moreover, allows to coarse-grain and reconstruct fluxes. Furthermore, we investigate the connection with functional inequalities and Poincaré-type constants.

In this paper, we prove that when a group $G$ acts on a tree $\Gamma$ such that the quotient graph $G/\Gamma$ is $path\ 2$ and when in $G$, any element of the stabilizer of one of the segments of this path commutes with those of the stabilizer of the other segment, $G$ may be identified with the free product of groups with commuting subgroups and every free product of groups with commuting subgroups is obtained uniquely in this way. Also, we give here some illustrative trees of this characterization.

Petri Nets (PN) are widely used for modeling concurrent and distributed systems, but face challenges in modeling adaptive systems. To address this, we have formalized "rewritable" PT nets (RwPT) using Maude, a declarative language with sound rewriting logic semantics. Recently, we introduced a modular approach that utilizes algebraic operators to construct large RwPT models. This technique employs composite node labeling to outline symmetries in hierarchical organization, preserved through net rewrites. Once stochastic parameters are added to the formalism, we present an automated process to derive a lumped CTMC from the quotient graph generated by an RwPT.

Network alignment, a foundational technique for cross-domain applications such as recommendation systems and knowledge fusion, faces significant challenges in balancing computational efficiency and alignment accuracy. Although graph neural networks (GNNs) effectively capture structural and semantic relationships, their computational intensity—stemming from multi-layer matrix operations and high memory consumption—severely limits scalability. To address these limitations, this paper proposes ENAMOR (Efficient Network AlignMent via Quotient gRaph), an unsupervised framework incorporating three key components: 1) multi-scale representation learning that hierarchically aggregates local and global structural patterns through GNN layers; 2) embedding-driven graph coarsening via hashing-based quotient graph construction, reducing computational complexity by 60–80% while preserving topological and attribute information; and 3) Matched Neighborhood Consistency (MNC) optimization, which iteratively refines alignment matrices by enforcing structural congruence constraints. Extensive experiments on three real-world datasets (Douban Online-Offline, Allmovie-Imdb, ACM-DBLP) demonstrate that ENAMOR achieves a 13.47–94.56% reduction in runtime compared to state-of-the-art methods, alongside improvements of 0.27–13.47 percentage points in mean average precision (MAP) and 0.15–8.17 percentage points in precision@5. The framework effectively balances efficiency and accuracy, providing a scalable solution for cross-network analysis tasks.

Molecular descriptors of symmetrically configured carbon nanocones via quotient graph technique

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

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.

Petri Nets (PN) are extensively used as a robust formalism to model concurrent and distributed systems; however, they encounter difficulties in accurately modeling adaptive systems. To address this issue, we defined rewritable PT nets (RwPT) using Maude, a declarative language that ensures consistent rewriting logic semantics. Recently, we proposed a modular approach that employs algebraic operators to build extensive RwPT models. This methodology uses composite node labeling to maintain hierarchical organization through net rewrites and has been shown to be effective. Once stochastic parameters are integrated into the formalism, we introduce an automated procedure to derive a lumped CTMC from the quotient graph generated by a modular RwPT model. To demonstrate the effectiveness of our method, we present a fault-tolerant manufacturing system as a case study.

Topological indices are real numbers that are presented as graph parameters introduced during studies conducted on the molecular graphs in chemistry and can describe some physical and chemical properties of molecules. Algebraic graph theory is a helpful tool in a range of chemistry domain. Because it helps to explain how the different symmetries of molecules and crystals affect their structure and dynamics it is a powerful theoretical approach for forecasting both the common and uncommon characteristics of molecules. In this article we compute the Wiener index, hyper wiener index, Harary index and hyper Harary index of Unitary Cayley Quotient Graph

Contagion probability in linear threshold model

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.

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.