Category Archives: Topology

Cycles, Rings, Strong rings

In chemical graph theory or chemical-related topology a distinction is made between a cycle, a ring, and a strong ring. It is relatively easy to define a cycle:

A cycle is a continuous path (along edges) in which the start and end vertex are the same.

So this means a cycle (also sometimes called a circuit) can be of any size, regardless if there are shorter (or also longer) paths to get back to the ‘home vertex’. We can also say that a cycle is a closed path.

What might be a little more difficult to understand is the formal definition of a ring:

A ring is a cycle that is not the sum of any two smaller cycles.

It is important to emphasize both attributes: two and smaller. The first conclusion is: Okey, in a graph or net there might be rings that are the sum of three, four, five (smaller) cycles, but a cycle is not a ring if it is the sum of exactly two (smaller) cycles.

In the following figure the graph of a cube is shown and some cycles (highlighted in red) within this graph:

Which of these cycles are rings? I think it is clear, that the two cycles in the top row are rings. Furthermore, the cycle in the middle in the bottom row is a ring, while the left and right cycle are not rings.  Why? Well, the 6-cycle on the left is the sum of two smaller 4-cycles. The 6-cycle in the middle is the sum of three(!) smaller cycles but not the sum of two smaller cycles. And the 8-cycle on the right can be considered as the sum of a 4-cycle and a 6-cycle (the latter can be formed as the sum of two smaller 4-cycles, the top-most and the central one):

Well, but shouldn’t it be possible to build the 6-cycle in the middle as the sum of a 4-cycle (top-most) and a 6-cycle (which itself be the sum of the central and right 4-cycles)? Yes, that is correct, but the definition states that the cycles of a sum of cycles have to be smaller! This is not the case here!

Interestingly, there is an alternative definition of a ring which is, at least for me, much more intuitive and comprehensible:

A ring is a cycle that contains along its path no potential short-cut (of the length of exactly one edge) to the ‘home vertex’.

It is easy to see that the cycle on the left has one short-cut and the cycle on the right even two (dashed edges):

So, what are now strong rings? They can be defined as follows:

Strong rings are rings that are not the sum of any number of smaller cycles.

From that definition It follows that only the two cycles in the top row are strong rings.

I would like to thank Michael Fischer (@ZeoliteMiFi) for helpful discussions.




Basic and derived nets

There is an ongoing discussion on the question how certain metal-organic frameworks (MOFs) should be deconstructed in the most appropriate way in order to obtain their underlying nets.

The majority of cases of uncertainties are dealing with different ways of deconstructing certain classes of polytopic linkers. For example, an elongated planar tetracarboxylate linker (~ C2h symmetry) can be regarded as one 4-coordinated vertex (square or rectangular local geometry) or as two 3-c vertices joined by an extra edge (two triangles in the same plane), see the following figure:


The most prominent example is MOF-505 [DOI: 10.1002/anie.200462787] (identical with NOTT-100 [DOI: 10.1002/anie.200601991]) based on the 4-c square-planar Cu paddle-wheel motif as inorganic secondary building units (SBU), and the question arises if the underlying net is nbo (cubic, 4-c uninodal) or fof (trigonal, (3,4)-c binodal):


Likewise tetratopic linkers with two pairs of connection points twisted by ~ 90° (~ D2d symmetry) can be regarded as one 4-c vertex ([distorted] tetrahedral local geometry) or as two 3-c vertices (two triangles joined by an extra edge and being perpendicular oriented to each other), as shown in the following figure:


One example is DUT-11 (DOI: 10.1002/ejic.201000415). Here, the underlying net can be described either as pts or sur:


There are arguments for and against a far-reaching compartimentalization of the linker, i.e. to explicitly take into account the additional branching points within the linker or not. It has been argued, for example, that tetratopic linkers should always regarded as a single 4-c node, because this would reflect its chemical nature more appropriately. However, there are at least two arguments against the single 4-c consideration: (i) the local geometry and the shape of the linker is represented very badly; (ii) a considerable amount of topological information would be lost, if two 3-c vertices are fused into one single 4-c vertex; if we take the example of the (4,4)-c net pts (“tetrahedra plus squares”) there are no less than six different topologies that can be derived by splitting either the 4-c tetrahedron into two 3-c vertices or the 4-c square into two 3-c vertices and there are no clear reasons why all these should be indiscriminately classified as their parent net pts, see the following figure:


(All nets are shown in their augmented versions – see here.)

What’s your opinion about that issue?

Augmentation of nets

An augmented net, which is indicated by an appendix -a to the three letter code of the basic net, is obtained from its underlying basic net by replacing all vertices with their respective coordination figure, i.e. polyhedron or polygon. For instance, all the vertices of the bcu net are replaced with cubes, and the (two different kinds of) 4-coordinated vertices of the pts net with squares and tetrahedra, respectively (see Figure below).


The first obvious advantage of such representations is that it is much easier to identify and to perceive the actual coordination environment of the vertices; not only the coordination number and type, but also their relative orientation to each other, so the local geometry of the vertices is explicitly included and visualized. In this sense it is also helpful in terms of a more intuitive or visual way of classification of nets, for instance, if we investigate all the possible nets that are assembled by a combination of certain node-types, say, tetrahedra and triangles, or trigonal prisms with squares and so on. However, augmentation is beneficial in particular in order to elucidate relationships between nets that are similar to each other: Note that the vertices of the augmented version of a given net have different coordination numbers, coordination sequences and vertex symbols; and these specifications – the topology – may be identical with another basic net that has been derived by the deconstruction of the chemical compound leading to this specific net description, for instance bcu-a is identical with the net of polycubane (pcb).