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Centrality as a Vertex Invariant (or 'Atom Descriptor')

EDIT: After some more tests, I now realise that this is not really as great a vertex label/descriptor as I thought it was. For example, see these four graphs on 7 vertices that fail to distinguish vertices properly:

The first one should have a central vertex in a different class than the other blue vertices. The green class in the second graph should be split, and same for the third graph. And so on.

So, in the last post I talked about the ideas of Randić et al for calculating the 'centrality' of vertices in a graph. Interestingly, the numbers calculated for each vertex act as a kind of equivalence class label or vertex invariant. This is similar in many ways to Morgan numbers (sorry, Egon's post doesn't actually explain them, but they are the sum of degrees across extended neighbourhoods).

For example, here is one of the examples from the previous post:

With the centrality matrix in the middle, and the 'label' made by sorting the row elements in descending order to the right of that. Finally, these labels are converted to more easily read alphabetic ones - classes in some sense ('a' = {0, 5} and 'b' = {1, 2, 3, 4}). These classes make sense, given that the middle vertices are in the same class, with the rest in another class.

Compare this to the graph with the same ORS of [6, 6, 5, 5, 5, 5]:

The graph here is nearly the same - with only the edge 1:3 missing - yet the labels don't distinguish between vertices {1, 3} and vertices {2, 4}. One possibility mentioned in the paper is to use combinations of descriptors (fairly common practice in cheminformatics, I suspect). The simplest one that occurs to me is just to add the degree of a vertex to the start of the label. That makes the label for {1, 3} into "1320000", distinguishing them from the one for {2, 4} which is "2320000".

Anyway, here is a picture of a number of pairs of graphs with the same ORS, colored by the label just described:

Note how some (but not all!) of these have the 'same' equivalence classes in different arrangements. Be aware that the colors may not be totally meaningful when compared between graphs. Code for this is here.


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