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    Symmetry properties of subdivision graphs

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    The subdivision graph S(Σ)S(\Sigma) of a graph Σ\Sigma is obtained from Σ\Sigma by `adding a vertex' in the middle of every edge of \Si. Various symmetry properties of §(Σ)\S(\Sigma) are studied. We prove that, for a connected graph Σ\Sigma, S(Σ)S(\Sigma) is locally ss-arc transitive if and only if Σ\Sigma is s+12\lceil\frac{s+1}{2}\rceil-arc transitive. The diameter of S(Σ)S(\Sigma) is 2d+δ2d+\delta, where Σ\Sigma has diameter dd and 0δ20\leqslant \delta\leqslant 2, and local ss-distance transitivity of §(Σ)\S(\Sigma) is defined for 1s2d+δ1\leqslant s\leqslant 2d+\delta. In the general case where s2d1s\leqslant 2d-1 we prove that S(Σ)S(\Sigma) is locally ss-distance transitive if and only if Σ\Sigma is s+12\lceil\frac{s+1}{2}\rceil-arc transitive. For the remaining values of ss, namely 2ds2d+δ2d\leqslant s\leqslant 2d+\delta, we classify the graphs Σ\Sigma for which S(Σ)S(\Sigma) is locally ss-distance transitive in the cases, s5s\leqslant 5 and s15+δs\geqslant 15+\delta. The cases max{2d,6}smin{2d+δ,14+δ}\max\{2d, 6\}\leqslant s\leqslant \min\{2d+\delta, 14+\delta\} remain open

    A semidefinite programming hierarchy for packing problems in discrete geometry

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    Packing problems in discrete geometry can be modeled as finding independent sets in infinite graphs where one is interested in independent sets which are as large as possible. For finite graphs one popular way to compute upper bounds for the maximal size of an independent set is to use Lasserre's semidefinite programming hierarchy. We generalize this approach to infinite graphs. For this we introduce topological packing graphs as an abstraction for infinite graphs coming from packing problems in discrete geometry. We show that our hierarchy converges to the independence number.Comment: (v2) 25 pages, revision based on suggestions by referee, accepted in Mathematical Programming Series B special issue on polynomial optimizatio
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