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The resistance distance between vertices i and j of a graph G is defined as the effective resistance between the two vertices (as when a battery is attached across them) when ...

A space-filling polyhedron is a polyhedron which can be used to generate a tessellation of space. Although even Aristotle himself proclaimed in his work On the Heavens that ...

Let Y_n denote the graph with vertex set V(X_n), where X_n is the n-hypercube and two vertices are adjacent in Y_n iff they are at distance 1<=d<=2 in X_n. Y_n is not ...

There are a number of algebraic equations known as the icosahedral equation, all of which derive from the projective geometry of the icosahedron. Consider an icosahedron ...

Nonstandard analysis is a branch of mathematical logic which introduces hyperreal numbers to allow for the existence of "genuine infinitesimals," which are numbers that are ...

An integer m such that if p|m, then p^2|m, is called a powerful number. There are an infinite number of powerful numbers, and the first few are 1, 4, 8, 9, 16, 25, 27, 32, ...

The Catalan numbers on nonnegative integers n are a set of numbers that arise in tree enumeration problems of the type, "In how many ways can a regular n-gon be divided into ...

A graph G is Hamilton-connected if every two vertices of G are connected by a Hamiltonian path (Bondy and Murty 1976, p. 61). In other words, a graph is Hamilton-connected if ...

A roulette is a curve traced by a fixed point on a closed convex curve as that curve rolls without slipping along a second curve. The roulettes described by the foci of ...

Let G be a group, and let S subset= G be a set of group elements such that the identity element I not in S. The Cayley graph associated with (G,S) is then defined as the ...