We show that the Hilbert bimodule associated with a compact topological graph can be recovered from the $C^*$-algebraic triple consisting of the Toeplitz algebra of the graph, its gauge action and the commutative subalgebra of functions on the vertex space of the graph. We discuss connections with work of Davidson–Katsoulis and of Davidson–Roydor on local conjugacy of topological graphs and isomorphism of their tensor algebras. In particular, we give a direct proof that a compact topological graph can be recovered up to local conjugacy from its Hilbert bimodule, and present an example of nonisomorphic locally conjugate compact topological graphs with isomorphic Hilbert bimodules. We also give an elementary proof that for compact topological graphs with totally disconnected vertex space the notions of local conjugacy, Hilbert bimodule isomorphism, isomorphism of $C^*$-algebraic triples, and isomorphism all coincide.

The research of parallel mechanism (PM) configuration involves many problems. From topology to configuration, dimensional constraint, etc., how to establish the relationship between topology and configuration with effective methods is a long-term challenge for the configuration design. In this paper, the chemical molecular spatial structure (CMSS) is linked with the configuration of symmetrical parallel mechanism (SPM). Starting from the methane molecule (CH4), a spatial structural topological relation is obtained. Based on graph theory and the spatial structural topological relation, a new expression method with topological graph and its kinematic pair adjacency matrix for spatial SPM is proposed. Then, the expression and analysis for the characteristics of spatial SPM are obtained. Finally, by taking the 3-RPS PM and the 3-RRC PM as examples, the effectiveness and corresponding consistency of the proposed expression method are successfully verified. The proposed new expression method paves the way for the subsequent digital and automated design and analysis of the SPM configuration.

Type synthesis of spatial 3-DoF (degree of freedom) parallel mechanisms (PMs) with planar sub-chains is studied using the revised digital topology graph (r-DTGs) and arrays. First, many DTGs for type synthesis of spatial 3-DoF PMs are derived from their contracted graphs (CGs). Second, a complicated derivation of the DTG and an identification of an isomorphic DTG are transformed into a simple derivation of an array and an identification of an isomorphic array using a compiled program. Third, one or more spatial closed-loop chains in the derived DTGs are changed into the planar closed-loop chains by modifying digits marked in closed-loop chains, and the DTGs are transformed into the r-DTGs for the type synthesis of the spatial 3-DoF PMs with planar sub-chains. Finally, the 52 novel spatial 3-DoF PMs with the planar sub-chains are synthesized and verified by simulation mechanisms.

A computational derivation of valid kinematic limbs for spatial 3-DOF parallel mechanisms (PMs) without redundant constraint is studied based on contracted graphs, topological graphs, and basic joints. First, some contracted graphs without any binary links are constructed, some curves with only binary links are distributed over contracted graphs and many valid topological graphs are derived. Second, a software is developed in Visual Basic for deriving the kinematic limb, and some basic chain structures are constructed by connecting various basic joints in series. Third, all valid kinematic limbs of the 3-DOF PMs without redundant constraint are derived computationally from the basic chain structures and some novel 3-DOF PMs are synthesized using this approach. Finally, the number of the different 3-DOF PMs without redundant constraint are determined based on the topological graphs and valid chain structures of the limb.