517 research outputs found
A Complete Characterisation of Vertex-multiplications of Trees with Diameter 5
For a connected graph , let be the family of strong orientations of ; and for any , we denote by the diameter of . The of is defined as . In 2000, Koh and Tay introduced a new family of graphs, vertex-multiplications, and extended the results on the orientation number of complete -partite graphs. Suppose has the vertex set . For any sequence of positive integers , a \textit{vertex-multiplication}, denoted by , is the graph with vertex set and edge set , where \u27s are pairwise disjoint sets with , for ; and for any , if and only if and for some with such that . They proved a fundamental classification of vertex-multiplications, with for all , into three classes and , and any vertex-multiplication of a tree with diameter at least 3 does not belong to the class . Furthermore, some necessary and sufficient conditions for were established for vertex-multiplications of trees with diameter . In this paper, we give a complete characterisation of vertex-multiplications of trees with diameter in and
Optimal orientations of Vertex-multiplications of Trees with Diameter 4
\noindent Koh and Tay proved a fundamental classification of vertex-multiplications into three classes and . They also showed that any vertex-multiplication of a tree with diameter at least 3 does not belong to the class . Of interest, vertex-multiplications are extensions of complete -partite graphs and Gutin characterised complete bipartite graphs with orientation number 3 (or 4 resp.) via an ingenious use of Sperner\u27s theorem. In this paper, we investigate vertex-multiplications of trees with diameter in (or ) and exhibit its intricate connections with problems in Sperner Theory, thereby extending Gutin\u27s approach. Let denote the vertex-multiplication of the central vertex. We almost completely characterise the case of even and give a complete characterisation for the case of odd
Improving the efficiency of variational tensor network algorithms
We present several results relating to the contraction of generic tensor
networks and discuss their application to the simulation of quantum many-body
systems using variational approaches based upon tensor network states. Given a
closed tensor network , we prove that if the environment of a
single tensor from the network can be evaluated with computational cost
, then the environment of any other tensor from can be
evaluated with identical cost . Moreover, we describe how the set of
all single tensor environments from can be simultaneously
evaluated with fixed cost . The usefulness of these results, which are
applicable to a variety of tensor network methods, is demonstrated for the
optimization of a Multi-scale Entanglement Renormalization Ansatz (MERA) for
the ground state of a 1D quantum system, where they are shown to substantially
reduce the computation time.Comment: 12 pages, 8 figures, RevTex 4.1, includes reference implementation.
Software updated to v1.02: Resolved two scenarios in which multienv would
generate errors for valid input
Bounds for the minimum oriented diameter
We consider the problem of finding an orientation with minimum diameter of a
connected bridgeless graph. Fomin et. al. discovered a relation between the
minimum oriented diameter an the size of a minimal dominating set. We improve
their upper bound.Comment: 21 pages, 6 figure
Hinged Dissections Exist
We prove that any finite collection of polygons of equal area has a common
hinged dissection. That is, for any such collection of polygons there exists a
chain of polygons hinged at vertices that can be folded in the plane
continuously without self-intersection to form any polygon in the collection.
This result settles the open problem about the existence of hinged dissections
between pairs of polygons that goes back implicitly to 1864 and has been
studied extensively in the past ten years. Our result generalizes and indeed
builds upon the result from 1814 that polygons have common dissections (without
hinges). We also extend our common dissection result to edge-hinged dissections
of solid 3D polyhedra that have a common (unhinged) dissection, as determined
by Dehn's 1900 solution to Hilbert's Third Problem. Our proofs are
constructive, giving explicit algorithms in all cases. For a constant number of
planar polygons, both the number of pieces and running time required by our
construction are pseudopolynomial. This bound is the best possible, even for
unhinged dissections. Hinged dissections have possible applications to
reconfigurable robotics, programmable matter, and nanomanufacturing.Comment: 22 pages, 14 figure
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