Labyrinth graph

A large number of different chemical classification systems exist for network topologies. These notably include those based on simple chemical compounds (e.g. diamondoid/cristabolite-type), an (n,p) system used by Wells related to that of Schlafli that classifies according to the number of links in a loop (n) and the node connectivity (p) [e.g. (6,4)]/ a three-letter system derived from that used for zeolites (e.g. dia,dia-a,dia-b,etc.and a 2D hyperbolic approach (e.g. sqc6). As an example, the chiral (10,3)-a network is known also as the SrSi2 net, srs. Laves net, Y, 3/10/cl, K4 crystal, and labyrinth graph of the G surface. 

Unfortunately there are no lUPAC or lUCr recommendations, or even a consensus among the scientists in the field, about the nomenclature of 3D-nets, and several naming systems are currently in use. As noted by O Keeffe et al. some have many names and symbols...other structures have no names at all and they give the example of the srs net that is also know as (10,3)-a , Laves net , Y , 3/10/cl , SrSi2 and labyrinth graph of the gyroid surface [1,2]. It can also be described by various sets of numbers, the most complete being lOs lOs lOs, referred to as the vertex symbol. Unfortunately, there is no present nomenclature that creates a set of numbers for each net that can be proven to be unique. 

Figure 4.4 Left The srs or (10,3)-a net also known as Laves net , Y , 3/IO/ct , SrSii and labyrinth graph of the gyroid surface" [1J. Right A 10-ring contained in this net.

Rucker, G. and Rucker, C. (2000) Walk cormts, labyrinthicity, and complexity of acyclic and cyclic graphs and molecules. /. Chem. Inf. Comput. Sci., 40, 99-106. 

Case (C) involves three (electron) densities, one for each labyrinth and one for the surface layer. Because relative intensities are all that are usually measured, only one parameter is needed to allow for any combination of three densities. The scattering function calculated in case (B) is frequently referred to as the structure factor of the surface for a minimal surface characterized by a self-dual skeletal graph, this will correspond to a different space group than that of (A) or (C), or of case (B) with nonzero mean curvature (see the D family, above). 

The portion of the volume inside the unit cell U for S that is part of the labyrinth containing the skeletal graph G is exactly described by the condition w < w, and w > w describes the G"-labyrinth, or ... 

The I-WP and F-RD minimal surfaces have been shown to provide two counterexamples to a conjecture that has previously been made (Meeks 1978, p. 81, Conjecture 6) A triply periodic minimal surface disconnects into two regions with asymptotically the same volume. The volume fraction of the labyrinth containing the symmetric skeletal graph is 0.5360 0.0002 for the I-WP minimal surface and 0.5319 0.0001 for F-RD. 

G. Rucker and C. Rucker, Walk counts, labyrinthicity and complexity of acyeUe and eyclic graphs and molecules, J. Chem. Inf. Comput. ScL 40 (2000) 99-106. 

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