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A set of positive integers is double-free if, for any integer x, the set {x,2x} !subset= S (or equivalently, x in S implies 2x not in S). For example, of the subsets of ...

An imperfect graph G is a graph that is not perfect. Therefore, graphs G with omega(G)<chi(G) (1) where omega(G) is the clique number and chi(G) is the chromatic number are ...

Two nonisomorphic graphs that have equal resistance spectra (i.e., multisets of resistance distances) are said to be resistance-equivalent. All nonisomorphic simple graphs on ...

There appears to be no term in standard use for a graph with graph crossing number 1. Furthermore, the terms "almost planar" and "1-planar" are used in the literature for ...

A reference triangle is a triangle relative to which trilinear coordinates, exact trilinear coordinates, derived triangle, or other objects are defined in triangle geometry ...

For every positive integer n, there exists a circle which contains exactly n lattice points in its interior. H. Steinhaus proved that for every positive integer n, there ...

A finite field is a field with a finite field order (i.e., number of elements), also called a Galois field. The order of a finite field is always a prime or a power of a ...

A Pythagorean triple is a triple of positive integers a, b, and c such that a right triangle exists with legs a,b and hypotenuse c. By the Pythagorean theorem, this is ...

A finite simple group of Lie-type. The following table summarizes the types of twisted Chevalley groups and their respective orders. In the table, q denotes a prime power and ...

The Wiener index W, denoted w (Wiener 1947) and also known as the "path number" or Wiener number (Plavšić et al. 1993), is a graph index defined for a graph on n nodes by ...