Showing papers on "Periodic graph (geometry) published in 2011"
TL;DR: Some continuous and discrete methods for solving some problems of molecular distance geometry involve a search in a continuous Euclidean space but sometimes the problem structure helps reduce the search to a discrete set of points.
112 citations
TL;DR: The existence for the perfect state transfer between antipodal vertices in graphs with extreme diameter is given.
Abstract: Let $G$ be a graph with adjacency matrix $A$, let $H(t)=\exp(itA)$. $G$ is called a periodic graph if there exists a time $\tau$ such that $H(\tau)$ is diagonal. If $u$ and $v$ are distinct vertices in $G$, we say that perfect state transfer occurs from $u$ to $v$ if there exists a time $\tau$ such that $|H(\tau)_{u,v}|=1$. A necessary and sufficient condition for $G$ is periodic is given. We give the existence for the perfect state transfer between antipodal vertices in graphs with extreme diameter.
18 citations
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TL;DR: This work discusses assumptions that bound the search tree width to a polynomial size and exploits the natural vertex order to show that the search space for feasible embeddings is discrete.
Abstract: An important application of distance geometry to biochemistry studies the embeddings of the vertices of a weighted graph in the three-dimensional Euclidean space such that the edge weights are equal to the Euclidean distances between corresponding point pairs. When the graph represents the backbone of a protein, one can exploit the natural vertex order to show that the search space for feasible embeddings is discrete. The corresponding decision problem can be solved using a binary tree based search procedure which is exponential in the worst case. We discuss assumptions that bound the search tree width to a polynomial size.
4 citations
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TL;DR: In this article, a refinement of the theorem of Bieberbach is presented, where the symmetry group of a uniformly discrete Euclidean graph has a subgroup isomorphic to its symmetry group.
Abstract: A uniformly discrete Euclidean graph is a graph embedded in a Euclidean space so that there is a minimum distance between distinct vertices. If such a graph embedded in an $n$-dimensional space is preserved under $n$ linearly independent translations, it is "$n$-periodic" in the sense that the quotient group of its symmetry group divided by the translational subgroup of its symmetry group is finite. We present a refinement of a theorem of Bieberbach: given a $n$-periodic uniformly discrete Euclidean graph embedded in a $n$-dimensional Euclidean space of symmetry group $\bbbS$, there is another $n$-periodic uniformly discrete Euclidean graph embedded in the same space whose vertices are integer points (possibly modulo an affine transformation) and whose symmetry group has a (not necessarily proper) subgroup isomorphic to $\bbbS$. We conclude with a discussion of an application to the computer generation of "crystal nets".
1 citations