1,837 research outputs found
Calculating the number of Hamilton cycles in layeredpolyhedral graphs
We describe a method for computing the number of Hamilton cycles in cubic polyhedral graphs. The Hamilton cycle counts are expressed in terms of a finite-state machine, and can be written as a matrix expression. In the special case of polyhedral graphs with repeating layers, the state machines become cyclic, greatly simplifying the expression for the exact Hamilton cycle counts, and let us calculate the exact Hamilton cycle counts for infinite series of graphs that are generated by repeating the layers. For some series, these reduce to closed form expressions, valid for the entire infinite series. When this is not possible, evaluating the number of Hamiltonian cycles admitted by the series' k-layer member is found by computing a (k - 1)th matrix power, requiring O(log(2)(k)) matrix-matrix multiplications. We demonstrate our technique for the two infinite series of fullerene nanotubes with the smallest caps. In addition to exact closed form and matrix expressions, we provide approximate exponential formulas for the number of Hamilton cycles.Peer reviewe
Computation of Contour Integrals on
Contour integrals of rational functions over , the moduli
space of -punctured spheres, have recently appeared at the core of the
tree-level S-matrix of massless particles in arbitrary dimensions. The contour
is determined by the critical points of a certain Morse function on . The integrand is a general rational function of the puncture
locations with poles of arbitrary order as two punctures coincide. In this note
we provide an algorithm for the analytic computation of any such integral. The
algorithm uses three ingredients: an operation we call general KLT, Petersen's
theorem applied to the existence of a 2-factor in any 4-regular graph and
Hamiltonian decompositions of certain 4-regular graphs. The procedure is
iterative and reduces the computation of a general integral to that of simple
building blocks. These are integrals which compute double-color-ordered partial
amplitudes in a bi-adjoint cubic scalar theory.Comment: 36+11 p
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