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Given a "good" graph G (i.e., one for which all intersecting graph edges intersect in a single point and arise from four distinct graph vertices), the crossing number is the ...

A product involving an infinite number of terms. Such products can converge. In fact, for positive a_n, the product product_(n=1)^(infty)a_n converges to a nonzero number iff ...

The Kirchhoff index Kf, also simply called the resistance and denoted R (Lukovits et al. 1999), of a connected graph G on n nodes is defined by ...

The Möbius strip, also called the twisted cylinder (Henle 1994, p. 110), is a one-sided nonorientable surface obtained by cutting a closed band into a single strip, giving ...

In its original form, the Poincaré conjecture states that every simply connected closed three-manifold is homeomorphic to the three-sphere (in a topologist's sense) S^3, ...

A series is an infinite ordered set of terms combined together by the addition operator. The term "infinite series" is sometimes used to emphasize the fact that series ...

A spanning tree of a graph on n vertices is a subset of n-1 edges that form a tree (Skiena 1990, p. 227). For example, the spanning trees of the cycle graph C_4, diamond ...

A sphere is defined as the set of all points in three-dimensional Euclidean space R^3 that are located at a distance r (the "radius") from a given point (the "center"). Twice ...

An (ordinary) torus is a surface having genus one, and therefore possessing a single "hole" (left figure). The single-holed "ring" torus is known in older literature as an ...

Geometry