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The Franklin graph is the 12-vertex cubic graph shown above whose embedding on the Klein bottle divides it into regions having a minimal coloring using six colors, thus ...

Define the abundancy Sigma(n) of a positive integer n as Sigma(n)=(sigma(n))/n, (1) where sigma(n) is the divisor function. Then a pair of distinct numbers (k,m) is a ...

The Gegenbauer polynomials C_n^((lambda))(x) are solutions to the Gegenbauer differential equation for integer n. They are generalizations of the associated Legendre ...

Turing machines are defined by sets of rules that operate on four parameters: (state, tape cell color, operation, state). Let the states and tape cell colors be numbered and ...

An algorithm for finding closed form hypergeometric identities. The algorithm treats sums whose successive terms have ratios which are rational functions. Not only does it ...

Consider the recurrence equation defined by a_0=m and a_n=|_sqrt(2a_(n-1)(a_(n-1)+1))_|, (1) where |_x_| is the floor function. Graham and Pollak actually defined a_1=m, but ...

The bandwidth of a connected graph G is the minimum matrix bandwidth among all possible adjacency matrices of graphs isomorphic to G. Equivalently, it is the minimum graph ...

There are several definitions of the strength of a graph. Harary and Palmer (1959) and Harary and Palmer (1973, p. 66) define the strength of a tree as the maximum number of ...

Grünbaum conjectured that for every m>1, n>2, there exists an m-regular, m-chromatic graph of girth at least n. This result is trivial for n=2 and m=2,3, but only a small ...

A Halin graph, sometimes known as a roofless polyhedron, is a polyhedral graph constructed from a planar embedding of a tree having four or more vertices, no vertices of ...