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Showing posts from October, 2013

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 ord…

Common Vertex Matrices of Graphs

There is an interesting set of papers out this year by Milan Randic et al (sorry about the accents - blogger seems to have a problem with accented 'c'...). I've looked at his work before here.

[1] Common vertex matrix: A novel characterization of molecular graphs by counting
[2] On the centrality of vertices of molecular graphs

and one still in publication to do with fullerenes. The central idea here (ho ho) is a graph descriptor a bit like path lengths called 'centrality'. Briefly, it is the count of neighbourhood intersections between pairs of vertices. Roughly this is illustrated here:

For the selected pair of vertices, the common vertices are those at the same distance from each - one at a distance of two and one at a distance of three. The matrix element for this pair will be the sum - 2 - and this is repeated for all pairs in the graph. Naturally, this is symmetric:

At the right of the matrix is the row sum (∑) which can be ordered to provide a graph invarian…