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The Hosoya index is an important topological index in graph theory, which is defined as the total number of s-matchings, denoted as m (G , s) , in a graph G. Therefore, computing the number of s-matchings for various molecular structures holds significant importance. By applying the concept of symmetry, defining the s-matching vector of the graph with a specified edge, using the transfer matrix, and iteratively applying two recursive formulas to derive the reduction formula, we compute the number of s-matchings of cyclooctatetraene chains. [ABSTRACT FROM AUTHOR]

Wireless mesh networks (WMNs) are a type of wireless network that can be used for various applications, such as Internet access, disaster response, and military communication. These networks consist of mesh routers that can communicate with each other and form a mesh topology, allowing them to provide connectivity even if some of the routers fail or are out of range. In this study, we used spectral graph theory to analyze the performance of a WMN. For the analysis process, software was developed to calculate the topological characteristics of the graph representing the WMN. The correlation between the values of the parameters of spectral graph theory and the topological characteristics of the observed network is analyzed. First, an analysis of the influence of the change in signal strength, in the observedWMN, on the algebraic connectivity was performed, and then the change in the spectral radius was also analyzed. The analysis was performed using special software, which was developed for that purpose. [ABSTRACT FROM AUTHOR]

We study the connection between multimatroids and moduli spaces of rational curves with cyclic action. Multimatroids are generalizations of matroids and delta-matroids that naturally arise in topological graph theory. The perspective of moduli of curves provides a tropical framework for studying multimatroids, generalizing the previous connection between type- |$A$| permutohedral varieties (Losev–Manin moduli spaces) and matroids, and the connection between type- |$B$| permutohedral varieties and delta-matroids. Specifically, we equate a combinatorial nef cone of the moduli space with the space of |${\mathbb {R}}$| -multimatroids, a generalization of multimatroids, and we introduce the independence polytopal complex of a multimatroid, whose volume is identified with an intersection number on the moduli space. As an application, we give a combinatorial formula for a natural class of intersection numbers on the moduli space by relating to the volumes of independence polytopal complexes of multimatroids. [ABSTRACT FROM AUTHOR]

In this paper. A new definition of discrete topological graph is introduced. Some properties of this graph are proved. If n > 2 are evaluated Gτ has no pendant vertex, not tree, also the value of the diameter and the minimum degree of Gτ . If n ≥ 2, Gτ has (2n − 3) complete bipartite induced subgraphs, Gτ is connected graph, simple graph, has no odd cycle, the clique number also proved, the value of the radius, the maximum degree and the chromatic number of Gτ have been studied. [ABSTRACT FROM AUTHOR]

In Topological graph theory, the maximum genus of graphs has been a fascinating subject. For a simple connected graph G, the maximum genus γM(G) is the largest genus of an orientable surface on which G has a 2-cell embedding. γM(G) has the upper bound ..., where β(G) denotes the Betti number and G is said to be upper embeddable if the equality holds. In this study, the maximum genus of GP(n, k) is established as ... for k = 1 and k = 2 by proving the upper embeddability of generalized Petersen graph, GP(n, k) for the cases k = 1 and k = 2. The proof is done by obtaining spanning trees T and examining the components in the edge complements GP(n, k)\T for the cases k = 1 and k = 2 of GP(n, k). [ABSTRACT FROM AUTHOR]

In line with the recent development in topological graph theory, we are considering undirected graphs that are allowed to contain multiple edges, loops, and semi-edges. A graph is called simple if it contains no semi-edges, no loops, and no multiple edges. A graph covering projection, also known as a locally bijective homomorphism, is a mapping between vertices and edges of two graphs which preserves incidences and which is a local bijection on the edge-neighborhood of every vertex. This notion stems from topological graph theory, but has also found applications in combinatorics and theoretical computer science. It has been known that for every fixed simple regular graph H of valency greater than 2, deciding if an input graph covers H is NP-complete. Graphs with semi-edges have been considered in this context only recently and only partial results on the complexity of covering such graphs are known so far. In this paper we consider the list version of the problem, called List-H-Cover, where the vertices and edges of the input graph come with lists of admissible targets. Our main result reads that the List-H-Cover problem is NP-complete for every regular graph H of valency greater than 2 which contains at least one semi-simple vertex (i.e., a vertex which is incident with no loops, with no multiple edges and with at most one semi-edge). Using this result we show the NP-co/polytime dichotomy for the computational complexity of List-H-Cover for cubic graphs. [ABSTRACT FROM AUTHOR]

We study the atomic embeddability testing problem, which is a common generalization of clustered planarity (c-planarity, for short) and thickenability testing, and present a polynomial-time algorithm for this problem, thereby giving the first polynomial-time algorithm for c-planarity. C-planarity was introduced in 1995 by Feng, Cohen, and Eades as a variant of graph planarity, in which the vertex set of the input graph is endowed with a hierarchical clustering and we seek an embedding (crossing free drawing) of the graph in the plane that respects the clustering in a certain natural sense. Until now, it has been an open problem whether c-planarity can be tested efficiently. The thickenability problem for simplicial complexes emerged in the topology of manifolds in the 1960s. A 2-dimensional simplicial complex is thickenable if it embeds in some orientable 3-dimensional manifold. Recently, Carmesin announced that thickenability can be tested in polynomial time. Our algorithm for atomic embeddability combines ideas from Carmesin's work with algorithmic tools previously developed for weak embeddability testing. We express our results purely in terms of graphs on surfaces, and rely on the machinery of topological graph theory. Finally, we give a polynomial-time reduction from atomic embeddability to thickenability thereby showing that both problems are polynomially equivalent, and showthat a slight generalization of atomic embeddability to the setting in which clusters are toroidal graphs is NP-complete. [ABSTRACT FROM AUTHOR]

Since the general definition of topology is based on the characteristics of the standard Euclidean topology, the relationships between the ordering on real numbers and its topology have been generalized over time and studied in numerous aspects. The compatibility of partially ordered sets with the topology on these sets was studied by many researchers. On the other hand, well-orderedness is an important concept of the set theory. We define the concept of topological well-orderedness, which can be regarded as a topological generalization of well-orderedness in the set theory and analyze its basic properties. In this way, the relationship between well-orderedness and topology is established from a different point of view. Finally, some basic applications of the concept of topological well-orderedness to the graph theory are investigated. [ABSTRACT FROM AUTHOR]

Other work of Euler would eventually become part of graph theory, e.g., the relation Graph HT ht connecting the number of vertices, faces, and edges of certain polyhedra and some lesser known polyhedra enumeration results [[9], Section 5]. The most revelatory experience of my undergraduate mathematics degree was a writing course offered by James Vick. The class readings included the recent Dover edition of Saaty and Kainan's book [[18]] on the "assaults and conquest" of that graph theory problem and Leonard Gillman's celebrated guide to mathematical writing [[5]]. But early graph theory was hardly dominated by American researchers, as evidenced by two "Graph Theory in Europe" interludes. [Extracted from the article]

A graph is said to be nearly complete bipartite if it can be obtained by deleting a set of independent edges from a complete bipartite graph. The nonorientable genus of such graphs is known except in a few cases where the sizes of the partite classes differ by at most one, and a maximum matching is deleted. We resolve these missing cases using three classic tools for constructing genus embeddings of the complete bipartite graphs: current graphs, diamond sums, and the direct rotation systems of Ringel. [ABSTRACT FROM AUTHOR]

The total structure connectivity and Narumi-Katayama indices of a simple graph G are defined as ... respectively, where du represents the degree of vertex u in G. In this paper, we determine the extremal values of total structure connectivity index on the class of unicyclic and bicyclic graphs and characterize the corresponding extremal graphs. In addition, we determine the bicyclic graphs extremal with respect to the Narumi-Katayama index. [ABSTRACT FROM AUTHOR]

A random two-cell embedding of a given graph $G$ is obtained by choosing a random local rotation around every vertex. We analyse the expected number of faces of such an embedding, which is equivalent to studying its average genus. In 1991, Stahl [5] proved that the expected number of faces in a random embedding of an arbitrary graph of order $n$ is at most $n\log (n)$. While there are many families of graphs whose expected number of faces is $\Theta (n)$ , none are known where the expected number would be super-linear. This led the authors of [1] to conjecture that there is a linear upper bound. In this note we confirm their conjecture by proving that for any $n$ -vertex multigraph, the expected number of faces in a random two-cell embedding is at most $2n\log (2\mu)$ , where $\mu$ is the maximum edge-multiplicity. This bound is best possible up to a constant factor. [ABSTRACT FROM AUTHOR]

The study of topological indices in graph theory is one of the more important topics, as the scientific development that occurred in the previous century had an important impact by linking it to many chemical and physical properties such as boiling point and melting point. So, our interest in this paper is to study many of the topological indices "generalized indices' network" for some graphs that have somewhat strange structure, so it is called the cog-graphs of special graphs "molecular network", by finding their polynomials based on vertex − edge degree then deriving them with respect to x , y , and x y , respectively, after substitution x = y = 1 of these special graphs are cog-path, cog-cycle, cog-star, cog-wheel, cog-fan, and cog-hand fan graphs; the importance of some types of these graphs is the fact that some vertices have degree four, which corresponds to the stability of some chemical compounds. These topological indices are first and second Zagreb, reduced first and second Zagreb, hyper Zagreb, forgotten, Albertson, and sigma indices. [ABSTRACT FROM AUTHOR]

Students' and graduates' mobility is an interesting topic of discussion especially for the Italian education system and universities. The main reasons for migration and for the so called brain drain, can be found in the socio-economic context and in the famous North–South divide. Measuring mobility and understanding its dynamic over time and space are not trivial tasks. Most of the studies in the related literature focus on the determinants of such phenomenon, in this paper, instead, combining tools coming from graph theory and Topological Data Analysis we propose a new measure for the attitude to mobility. Each mobility trajectory is represented by a graph and the importance of the features constituting the graph are evaluated over time using persistence diagrams. The attitude to mobility of the students is then ranked computing the distance between the individual persistence diagram and the theoretical persistence diagram of the stayer student. The new approach is used for evaluating the mobility of the students that in 2008 enrolled in an Italian university. The relation between attitude to mobility and the main socio-demographic variables is investigated. [ABSTRACT FROM AUTHOR]

Motivated by the problem arising out of DNA origami, we give a general counting framework and enumeration formulas for various cellular embeddings of bouquets and dipoles under different kinds of symmetries. Our algebraic framework can be used constructively to generate desired symmetry classes, and we use Burnside's lemma with various symmetry groups to derive the enumeration formulas. Our results assimilate several existing formulas into this unified framework. Furthermore, we provide new formulas for bouquets with colored edges (and thus for bouquets in nonorientable surfaces) as well as for directed embeddings of directed bouquets. We also enumerate vertex-labeled dipole embeddings. Since dipole embeddings may be represented by permutations, the formulas also apply to certain equivalence classes of permutations and permutation matrices. The resulting bouquet and dipole symmetry formulas enumerate structures relevant to a wide variety of areas in addition to DNA origami, including RNA secondary structures, Feynman diagrams, and topological graph theory. For uncolored objects, we catalog 58 distinct sequences, of which 43 have not, as far as we know, been described previously. [ABSTRACT FROM AUTHOR]

West syndrome is an epileptic disease that seriously affects the normal growth and development of infants in early childhood. Based on the methods of brain topological network and graph theory, this article focuses on three clinical states of patients before and after treatment. In addition to discussing bidirectional and unidirectional global networks from the perspective of computational principles, a more in-depth analysis of local intra-network and inter-network characteristics of multi-partitioned networks is also performed. The spatial feature distribution based on feature path length is introduced for the first time. The results show that the bidirectional network has better significant differentiation. The rhythmic feature change trend and spatial characteristic distribution of this network can be used as a measure of the impact on global information processing in the brain after treatment. And localized brain regions variability in features and differences in the ability to interact with information between brain regions have potential as biomarkers for medication assessment in WEST syndrome. The above shows specific conclusions on the interaction relationship and consistency of macro-network and micro-network, which may have a positive effect on patients' treatment and prognosis management. • Evaluate the brain network changes in three clinical states of patients before and after treatment. • Construct multi-partition local network for intra-network and inter-network feature analysis. • The bidirectional network has more significant differences than the unidirectional network. • Localized brain regions variability is effective for medication assessment in WEST syndrome. [ABSTRACT FROM AUTHOR]

Modeling a road network as a planar graph seems very natural. However, in studying continuum limits of such networks it is useful to take routes rather than edges as primitives. This article is intended to introduce the relevant (discrete setting) notion of routed network to graph theorists. We give a naive classification of all 71 topologically different such networks on 4 leaves, and pose a variety of challenging research questions.

The number Z (n) : = ⌊ n / 2 ⌋ ⌊ (n - 1) / 2 ⌋ is the smallest number of crossings in a simple planar drawing of K 2 , n in which both vertices on the 2-side have the same clockwise rotation. For two vertices u, v on the q-side of a simple drawing of K p , q , let cr D (u , v) denote the total number of crossings that edges incident with u have with edges incident with v. We show that in any simple drawing D of K p , q in a surface Σ the number of pairs of vertices on the q-side of K p , q having cr D (u , v) < Z (p) is bounded as a function of p and Σ . As a consequence, we also show that, for a fixed integer p and surface Σ , there exists a finite set of drawings D (p , Σ) of complete bipartite graphs such that, for each q, a crossing-minimal drawing of K p , q can be obtained by "duplicating vertices" in some drawing from D (p , Σ) . [ABSTRACT FROM AUTHOR]

Our purpose is to show that complements of line graphs (of graphs) enjoy nice coloring properties. We show that for all graphs in this class the local and usual chromatic numbers are equal. We also prove a sufficient condition for the chromatic number to be equal to a natural upper bound. A consequence of this latter condition is a complete characterization of all induced subgraphs of the Kneser graph KG(n,2) that have a chromatic number equal to its chromatic number, namely n−2. In addition to the upper bound, a lower bound is provided by Dol'nikov's theorem, a classical result of the topological method in graph theory. We prove the NP‐hardness of deciding the equality between the chromatic number and any of these bounds. The topological method is especially suitable for the study of coloring properties of complements of line graphs of hypergraphs. Nevertheless, all proofs in this article are elementary and we also provide a short discussion on the ability for the topological methods to cover some of our results. [ABSTRACT FROM AUTHOR]

A longstanding open question of Archdeacon and Craft asks whether every complete graph has a minimum genus embedding with at most one nontriangular face. We exhibit such an embedding for each complete graph except for K8, and we go on to prove that no such embedding can exist for this graph. Our approach also solves a more general problem, giving a complete characterization of the possible face distributions (i.e., the numbers of faces of each length) realizable by minimum genus embeddings of each complete graph. We also tackle analogous questions for nonorientable and maximum genus embeddings. [ABSTRACT FROM AUTHOR]

We show that the complete graphs on 24 s + 21 vertices have decompositions into two edge-disjoint subgraphs, each of which triangulates an orientable surface. The special case where the two surfaces are homeomorphic solves a generalized Earth-Moon problem for that surface. Unlike previous constructions, these pairs of triangular embeddings are derived from index 3 current graphs. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity and I ∗ (R) , the set of all nontrivial proper ideals of R. The intersection graph of ideals of R , denoted by ℐ (R) , is a simple undirected graph with vertex set as the set I ∗ (R) , and, for any two distinct vertices I and J are adjacent if and only if I ∩ J ≠ (0). In this paper, we study some connections between commutative ring theory and graph theory by investigating topological properties of intersection graph of ideals. In particular, it is shown that for any nonlocal Artinian ring R , ℐ (R) is a projective graph if and only if R ≅ R 1 × F 1 , where R 1 is a local principal ideal ring with maximal ideal 𝔪 1 of nilpotency three and F 1 is a field. Furthermore, it is shown that for an Artinian ring R , γ ¯ (ℐ (R)) = 2 if and only if R ≅ R 1 × R 2 , where each R i (1 ≤ i ≤ 2) is a local principal ideal ring with maximal ideal 𝔪 i ≠ (0) such that 𝔪 i 2 = (0). [ABSTRACT FROM AUTHOR]

Introduction: Generalized tonic‐clonic seizure (GTCS) is a condition that is characterized by generalized spike‐wave discharge in bilateral cerebral hemispheres during the seizure. Although previous neuroimaging studies revealed functional abnormalities in the brain activities of children with GTCS, the topological alterations in whole‐brain networks remain poorly understood. Methods: The present study used graph theory to investigate the topological organization of functional networks in 13 GTCS children and 30 age‐matched healthy controls. Results: We found that both groups exhibited a small‐world topology of the functional network. However, children with GTCS showed a significant decrease in nodal local efficiency and clustering coefficient in some key nodes compared with the controls. The connections within the default mode network (DMN) were decreased significantly, and the internetwork connections were increased significantly. The altered topological properties may be an effect of chronic epilepsy. As a result, the optimal topological organization of the functional network was disrupted in the patient group. Notably, clustering coefficient and nodal local efficiency in the bilateral temporal pole of the middle temporal gyrus negatively correlated with the epilepsy duration. Conclusion: These results suggest that the bilateral temporal pole plays an important role in reflecting the effect of chronic epilepsy on the topological properties in GTCS children. The present study demonstrated a disrupted topological organization in children with GTCS. These findings provide new insight into our understanding of this disorder. [ABSTRACT FROM AUTHOR]

Graph theory is one of the rising areas in mathematics due to its applications in many areas of science. Amongst several study areas in graph theory, spectral graph theory and topological descriptors are in front rows. These descriptors are widely used in QSPR/QSAR studies in mathematical chemistry. Vertex-semitotal graphs are one of the derived graph classes which are useful in calculating several physico-chemical properties of molecular structures by means of molecular graphs modelling the molecules. In this paper, several topological descriptors of vertex- semitotal graphs are calculated. Some new relations on these values are obtained by means of a recently defined graph invariant called omega invariant. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce a computational tool that checks theoretical conditions in order to determine whether a weighted graph, as a topological invariant of stable maps, can be associated to stable maps without cusps (ie, fold maps) from closed surfaces to the projective plan. [ABSTRACT FROM AUTHOR]

A list of regular papers published in the July 2020 issue of "IEEE Transactions on Vehicular Technology" is presented, including "Impact of Big Vehicle Shadowing on Vehicle-to-Vehicle Communications," "Demand-Aware Provisioning of Electric Vehicles Fast Charging Infrastructure" and "Adaptive Switching Spatial-Temporal Fusion Detection for Remote Flying Drones."

Calculating the genus distributions of ladder graphs is a concerned topic in topological graph theory. In this paper, we formulate several ladder-class graphs by using a starting graph iterative amalgamation with copies of a path to construct a base graph and then adding some edges to the appointed root-vertices of the base graph. By means of transfer matrix and a finer partition of the embeddings, the explicit formulas for the genus distribution polynomials of four types of ladder-class graphs are derived. [ABSTRACT FROM AUTHOR]

The conferences in the series of the Japan Conference on Discrete and Computational Geometry, Graphs and Games ( JCDCG 3 ) have been held annually since 1997, except for 2008. Since 1997, almost one thousand research results have been presented in total at the conferences, and 11 post-conference proceedings and 6 special issues of journals have been published. To celebrate the 20th Anniversary of JCDCG 3 , a summary of the notable results published in those proceedings are presented in this article. We focus on six areas such as games and puzzles, dissection and reversibility, foldings and unfoldings, point sets, visibility, and geometric and topological graph theory. [ABSTRACT FROM AUTHOR]

Enumerating the isomorphism or equivalence classes of several types of graph coverings is one of the central research topics in enumerative topological graph theory. A covering projection p from a Cayley graph Cay(Γ, X) onto another Cayley graph Cay(Q, Y) is called typical if the function p : Γ → Q on the vertex sets is a group epimorphism. A typical covering is called abelian (or circulant, respectively) if its covering graph is a Cayley graph on an abelian (or a cyclic, respectively) group. Recently, the equivalence classes of connected abelian typical prime-fold coverings of a circulant graph are enumerated. As a continuation of this work, we enumerate the equivalence classes of connected abelian typical cube-free fold coverings of a circulant graph. [ABSTRACT FROM AUTHOR]

Mild cognitive impairment (MCI) is often considered a critical time window for predicting early conversion to Alzheimer's disease (AD). Brain functional connectome data (i.e., functional connections, global and nodal graph metrics) based on resting-state functional magnetic resonance imaging (rs-fMRI) provides numerous information about brain networks and has been used to discriminate normal controls (NCs) from subjects with MCI. In this paper, Student's t -tests and group-least absolute shrinkage and selection operator (group-LASSO) were used to extract functional connections with significant differences and the most discriminative network nodes, respectively. Based on group-LASSO, the middle temporal, inferior temporal, lingual, posterior cingulate, and middle frontal gyri were the most predominant brain regions for nodal observation in MCI patients. Nodal graph metrics (within-module degree, participation coefficient, and degree centrality) showed the maximum discriminative ability. To effectively combine the multipattern information, we employed the multiple kernel learning support vector machine (MKL-SVM). Combined with functional connectome information, the MKL-SVM achieved a good classification performance (area under the receiving operating characteristic curve = 0.9728). Additionally, the altered brain connectome pattern revealed that functional connectivity was generally decreased in the whole-brain network, whereas graph theory topological attributes of some special nodes in the brain network were increased in MCI patients. Our findings demonstrate that optimal feature selection and combination of all connectome features (i.e., functional connections, global and nodal graph metrics) can achieve good performance in discriminating NCs from MCI subjects. Thus, the combination of functional connections and global and nodal graph metrics of brain networks can predict the occurrence of MCI and contribute to the early clinical diagnosis of AD. [ABSTRACT FROM AUTHOR]

The purpose of the present work was to study the real life problems using neutrosophic nano topological graph theory. Most real-life situations need some sort of approximation to fit mathematical models. The beauty of using neutrosophic nano topology in approximation is achieved via approximation for qualitative sub graphs without coding or using assumption. By certain nano equivalence relation, we are formalizing the structural equivalence of basic circuit of the LED light from the graphs and their corresponding neutrosophic nano topologies generated by them. [ABSTRACT FROM AUTHOR]

Identifying graphs with extremal properties is an extensively studied topic in both topological graph theory and spectral graph theory. As observed in the literature, for many graph categories the extremal graphs with respect to some prescribed topological index or spectral invariant are the same. Therefore, it is an interesting problem to find a unified method to identify the extremal graphs for a set of topological or spectral invariants. Here we consider the majorization theorem to obtain extremal graphs in the class of trees, unicyclic graphs and bicyclic graphs with given order and fixed maximum degree, independence number, matching number, domination number, and/or number of pendant vertices, respectively. [ABSTRACT FROM AUTHOR]

Acupuncture manipulation is the key of Chinese medicine acupuncture therapy. In clinical practice, different acupuncture manipulations are required to achieve different therapeutic effects, which means it is crucial to distinguish different acupuncture manipulations. In this paper, we proposed a classification framework for different acupuncture manipulations, which employed the graph theory and machine learning method. Multichannel EEG signals evoked by acupuncture at “Zusanli” acupoint were recorded from healthy humans by two acupuncture manipulations: twirling-rotating (TR) and lifting-thrusting (LT). Phase locking value was used to estimate the phase synchronization of pair-wise EEG channels. It was found that acupunctured by TR manipulation exhibit significantly higher synchronization degree than acupunctured by LT manipulation. With the construction of functional brain network, the topological features of graph theory were extracted. Taken the network features as inputs, machine learning classifiers were established to classify acupuncture manipulations. The highest accuracy can achieve 92.14% with support vector machine. By further optimizing the network features utilized in machine learning classifiers, it was found that the combination of node betweenness and small world network index is the most effective factor for acupuncture manipulations classification. These findings suggested that our approach provides new ideas for automatically identify acupuncture manipulations from the perspective of functional brain networks and machine learning methods. [ABSTRACT FROM AUTHOR]

• Spatial mechanism topological graph is obtained by evolving molecular structure. • The limitation that graph theory can only design planar mechanisms is removed. • A design method for RHM configuration synthesis is proposed. • A new class of RHM with large extension characteristics is obtained. • To realize high-precision machining of large complex curved surfaces in the future. A design method for hybrid mechanism configuration synthesis based on topological graph theory is proposed to address the challenge of machining large and complex curved surfaces in the aerospace domain. The method leads to the generation of a new class of reconfigurable large extension hybrid mechanisms. Firstly, the spatial mechanism topological graph (SMTG) is obtained by evolving the structure of chemical molecules, which is used to express the spatial mechanism topological relationships. Then, combining graph theory methods with the SMTG, the topological relationships between motion modules of the hybrid mechanism are expressed, and the mechanisms of each motion module are designed. Finally, mechanisms of different functional modules are topologically connected and combined using the proposed topological connection relationship for the motion modules of hybrid mechanisms, thus resulting in a class of reconfigurable hybrid mechanisms (RHMs) with large extensibility. The RHMs exhibit strong structural stability, a large working space, and a small volume of occupation. It can be installed on the ends of industrial robots or in large guideways in the future to achieve high-precision machining of large complex curved structures. [ABSTRACT FROM AUTHOR]