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Symmetry operations include the improper rotation, inversion operation, mirror plane, and rotation. Together, these operations create 32 crystal classes corresponding to the ...

Given two additive groups (or rings, or modules, or vector spaces) A and B, the map f:A-->B such that f(a)=0 for all a in A is called the zero map. It is a homomorphism in ...

An International Standard Book Number (ISBN) is a code used to uniquely identify a book together. It also uniquely encodes the book's publisher and includes information about ...

A multiplication table is an array showing the result of applying a binary operator to elements of a given set S. For example, the following table is the multiplication table ...

The space groups in two dimensions are called wallpaper groups. In three dimensions, the space groups are the symmetry groups possible in a crystal lattice with the ...

A stochastic matrix, also called a probability matrix, probability transition matrix, transition matrix, substitution matrix, or Markov matrix, is matrix used to characterize ...

A (0,2)-graph is a connected graph such that any two vertices have 0 or 2 common neighbors. (0,2)-graphs are regular, and the numbers of (0,2)-graphs with vertex degree 0, 1, ...

A chordal graph is a simple graph in which every graph cycle of length four and greater has a cycle chord. In other words, a chordal graph is a graph possessing no chordless ...

Let P(G) denote the chromatic polynomial of a finite simple graph G. Then G is said to be chromatically unique if P(G)=P(H) implies that G and H are isomorphic graphs, in ...

The (upper) clique number of a graph G, denoted omega(G), is the number of vertices in a maximum clique of G. Equivalently, it is the size of a largest clique or maximal ...