Tangled piecewise-linear embeddings of trivalent graphs

Quotient Complex (QC)-Based Machine Learning for 2D Hybrid Perovskite Design

With remarkable stability and exceptional optoelectronic properties, two-dimensional (2D) halide layered perovskites hold immense promise for revolutionizing photovoltaic technology. Effective data representations are key to the success of all learning models. Currently, the lack of comprehensive and accurate material representations has hindered AI-based design and discovery of 2D perovskites, limiting their potential for advanced photovoltaic applications. In this context, this work introduces a novel computational topology framework termed the quotient complex (QC), which serves as the foundation for the material representation. The proposed QC-based features are seamlessly integrated with learning models for the advancement of 2D perovskite design. At the heart of this framework lies the quotient complex descriptors (QCDs), representing a quotient variation of simplicial complexes derived from materials' unit cell and periodic boundary conditions. Differing from prior material representations, this approach encodes higher-order interactions and periodicity information simultaneously. Based on the well-established new materials for solar energetics (NMSE) databank, the proposed QC-based machine learning models exhibit superior performance against all existing counterparts. This underscores the paramount role of periodicity information in predicting material functionality, while also showcasing the remarkable efficiency of the QC-based model in characterizing materials' structural attributes.

Piecewise-linear embeddings of decussate extended θ graphs and tetrahedra

An nθ graph is an n-valent graph with two vertices. From symmetry considerations, it has vertex–edge transitivity 1 1. Here, they are considered extended with divalent vertices added to the edges to explore the simplest piecewise-linear tangled embeddings with straight, non-intersecting edges (sticks). The simplest tangles found are those with 3n sticks, transitivity 2 2, and with 2⌊(n − 1)/2⌋ ambient-anisotopic tangles. The simplest finite and 1-, 2- and 3-periodic decussate structures (links and tangles) are described. These include finite cubic and icosahedral and 1- and 3-periodic links, all with minimal transitivity. The paper also presents the simplest tangles of extended tetrahedra and their linkages to form periodic polycatenanes. A vertex- and edge-transitive embedding of a tangled srs net with tangled and polycatenated θ graphs and vertex-transitive tangled diamond (dia) nets are described.

The Symmetry and Topology of Finite and Periodic Graphs and Their Embeddings in Three-Dimensional Euclidean Space

We make the case for the universal use of the Hermann-Mauguin (international) notation for the description of rigid-body symmetries in Euclidean space. We emphasize the importance of distinguishing between graphs and their embeddings and provide examples of 0-, 1-, 2-, and 3-periodic structures. Embeddings of graphs are given as piecewise linear with finite, non-intersecting edges. We call attention to problems of conflicting terminology when disciplines such as materials chemistry and mathematics collide.