Metal–organic frameworkMetal–organic frameworks (MOFs) are a class of compounds consisting of metal clusters (also known as SBUs) coordinated to organic ligands to form one-, two-, or three-dimensional structures. The organic ligands included are sometimes referred to as "struts" or "linkers", one example being 1,4-benzenedicarboxylic acid (BDC). More formally, a metal–organic framework is an organic-inorganic porous extended structure. An extended structure is a structure whose sub-units occur in a constant ratio and are arranged in a repeating pattern.
Graph theoryIn mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called nodes or points) which are connected by edges (also called links or lines). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics.
Graph coloringIn graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color; this is called a vertex coloring. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color.
Component (graph theory)In graph theory, a component of an undirected graph is a connected subgraph that is not part of any larger connected subgraph. The components of any graph partition its vertices into disjoint sets, and are the induced subgraphs of those sets. A graph that is itself connected has exactly one component, consisting of the whole graph. Components are sometimes called connected components. The number of components in a given graph is an important graph invariant, and is closely related to invariants of matroids, topological spaces, and matrices.
Crystal structureIn crystallography, crystal structure is a description of the ordered arrangement of atoms, ions, or molecules in a crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric patterns that repeat along the principal directions of three-dimensional space in matter. The smallest group of particles in the material that constitutes this repeating pattern is the unit cell of the structure.
CrystalA crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macroscopic single crystals are usually identifiable by their geometrical shape, consisting of flat faces with specific, characteristic orientations. The scientific study of crystals and crystal formation is known as crystallography.
Liquid crystalLiquid crystal (LC) is a state of matter whose properties are between those of conventional liquids and those of solid crystals. For example, a liquid crystal may flow like a liquid, but its molecules may be oriented in a crystal-like way. There are many types of LC phases, which can be distinguished by their optical properties (such as textures). The contrasting textures arise due to molecules within one area of material ("domain") being oriented in the same direction but different areas having different orientations.
Covalent organic frameworkCovalent organic frameworks (COFs) are a class of materials that form two- or three-dimensional structures through reactions between organic precursors resulting in strong, covalent bonds to afford porous, stable, and crystalline materials. COFs emerged as a field from the overarching domain of organic materials as researchers optimized both synthetic control and precursor selection.
Graph rewritingIn computer science, graph transformation, or graph rewriting, concerns the technique of creating a new graph out of an original graph algorithmically. It has numerous applications, ranging from software engineering (software construction and also software verification) to layout algorithms and picture generation. Graph transformations can be used as a computation abstraction. The basic idea is that if the state of a computation can be represented as a graph, further steps in that computation can then be represented as transformation rules on that graph.
Graph homomorphismIn the mathematical field of graph theory, a graph homomorphism is a mapping between two graphs that respects their structure. More concretely, it is a function between the vertex sets of two graphs that maps adjacent vertices to adjacent vertices. Homomorphisms generalize various notions of graph colorings and allow the expression of an important class of constraint satisfaction problems, such as certain scheduling or frequency assignment problems.
Single crystalIn materials science, a single crystal (or single-crystal solid or monocrystalline solid) is a material in which the crystal lattice of the entire sample is continuous and unbroken to the edges of the sample, with no grain boundaries. The absence of the defects associated with grain boundaries can give monocrystals unique properties, particularly mechanical, optical and electrical, which can also be anisotropic, depending on the type of crystallographic structure.
Extremal graph theoryExtremal graph theory is a branch of combinatorics, itself an area of mathematics, that lies at the intersection of extremal combinatorics and graph theory. In essence, extremal graph theory studies how global properties of a graph influence local substructure.
Colloidal crystalA colloidal crystal is an ordered array of colloidal particles and fine grained materials analogous to a standard crystal whose repeating subunits are atoms or molecules. A natural example of this phenomenon can be found in the gem opal, where spheres of silica assume a close-packed locally periodic structure under moderate compression. Bulk properties of a colloidal crystal depend on composition, particle size, packing arrangement, and degree of regularity.
Random graphIn mathematics, random graph is the general term to refer to probability distributions over graphs. Random graphs may be described simply by a probability distribution, or by a random process which generates them. The theory of random graphs lies at the intersection between graph theory and probability theory. From a mathematical perspective, random graphs are used to answer questions about the properties of typical graphs.
ZeoliteZeolites are microporous, crystalline aluminosilicate materials commonly used as commercial adsorbents and catalysts. They mainly consist of silicon, aluminium, oxygen, and have the general formula M1/nn+(AlO2)-(SiO2)x・yH2O where M1/nn+ is either a metal ion or H+. These positive ions can be exchanged for others in a contacting electrolyte solution. H+ exchanged zeolites are particularly useful as solid acid catalysts.
Algebraic graph theoryAlgebraic graph theory is a branch of mathematics in which algebraic methods are applied to problems about graphs. This is in contrast to geometric, combinatoric, or algorithmic approaches. There are three main branches of algebraic graph theory, involving the use of linear algebra, the use of group theory, and the study of graph invariants. The first branch of algebraic graph theory involves the study of graphs in connection with linear algebra.
Materials scienceMaterials science is an interdisciplinary field of researching and discovering materials. Materials engineering is an engineering field of finding uses for materials in other fields and industries. The intellectual origins of materials science stem from the Age of Enlightenment, when researchers began to use analytical thinking from chemistry, physics, and engineering to understand ancient, phenomenological observations in metallurgy and mineralogy. Materials science still incorporates elements of physics, chemistry, and engineering.
Degeneracy (graph theory)In graph theory, a k-degenerate graph is an undirected graph in which every subgraph has a vertex of degree at most k: that is, some vertex in the subgraph touches k or fewer of the subgraph's edges. The degeneracy of a graph is the smallest value of k for which it is k-degenerate. The degeneracy of a graph is a measure of how sparse it is, and is within a constant factor of other sparsity measures such as the arboricity of a graph.
Crystallographic databaseA crystallographic database is a database specifically designed to store information about the structure of molecules and crystals. Crystals are solids having, in all three dimensions of space, a regularly repeating arrangement of atoms, ions, or molecules. They are characterized by symmetry, morphology, and directionally dependent physical properties. A crystal structure describes the arrangement of atoms, ions, or molecules in a crystal.
CrystallographyCrystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The word crystallography is derived from the Ancient Greek word κρύσταλλος (; "clear ice, rock-crystal"), with its meaning extending to all solids with some degree of transparency, and γράφειν (; "to write").
