Overhanging of membranes and filaments adhering to periodic graph substrates

This paper presents the benefits of using the random-walk normalized Laplacian matrix as a graph-shift operator and defines the frequencies of a graph by the eigenvalues of this matrix. A criterion to order these frequencies is proposed based on the Euclidean distance between a graph signal and its shifted version with the transition matrix as shift operator. Further, the frequencies of a periodic graph built through the repeated concatenation of a basic graph are studied. We show that when a graph is replicated, the graph frequency domain is interpolated by an upsampling factor equal to the number of replicas of the basic graph, similarly to the effect of zero-padding in digital signal processing.

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Crystal Structures of Inorganic Oxoacid Salts Perceived as Cation Arrays: A Periodic-Graph Approach

Inorganic 3D Structures - Structure and Bonding ◽

10.1007/430_2010_34 ◽

2010 ◽

pp. 31-66 ◽

Cited By ~ 18

Author(s):

Vladislav A. Blatov

Keyword(s):

Crystal Structures ◽

Periodic Graph

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Periodic Rigidity on a Variable Torus Using Inductive Constructions

The Electronic Journal of Combinatorics ◽

10.37236/2212 ◽

2015 ◽

Vol 22 (1) ◽

Cited By ~ 2

Author(s):

Anthony Nixon ◽

Elissa Ross

Keyword(s):

Rigidity Theory ◽

Periodic Graph ◽

Gain Graphs ◽

Generic Rigidity ◽

Periodic Frameworks

In this paper we prove a recursive characterisation of generic rigidity for frameworks periodic with respect to a partially variable lattice. We follow the approach of modelling periodic frameworks as frameworks on a torus and use the language of gain graphs for the finite counterpart of a periodic graph. In this setting we employ variants of the Henneberg operations used frequently in rigidity theory.

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Automatically generated periodic graphs

Zeitschrift für Kristallographie - Crystalline Materials ◽

10.1515/zkri-2015-1866 ◽

2015 ◽

Vol 230 (12) ◽

Author(s):

Gregory McColm

Keyword(s):

Crystal Structure ◽

Finite Graph ◽

Automatic Generation ◽

Unit Cell ◽

Affine Transformations ◽

Periodic Graphs ◽

Periodic Graph ◽

Crystal Design

AbstractA crystal structure may be described by a periodic graph, so the automatic generation of periodic graphs satisfying given criteria should prove useful in understanding crystal structure and developing a theory for crystal design. We present an algorithm for constructing periodic graphs, based on a discrete variant of “turtle geometry,” which we formalize using isometries represented as affine transformations. This algorithm is given information about the putative orbits (kinds) of vertices and edges and produces a finite graph which is then collapsed into a unit cell of a periodic graph. We verify that this algorithm will generate any periodic graph.

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