I am not familiar with studies about such indices, but will provide my ideas.

It is not easy to formalize mathematically level of polycentricity. The first is spatial separation. If there are maps with building areas and/or local population density, it is possible to conclude (just visually) if a certain city is polycentric or not.

Fernando, you are providing interesting pictures about the pattern of commuting with job places in a city. In general, it can be viewed as a network of individuals linked with certain attracting centers. Clearly, full information about the topology of such network is rarely available.

One may start to construct own index depending on available data and question to be answered. If the question is - how many centers offering substantial employment in a city and whether they are located in one or several districts - then your pictures can answer this question.

The figure is due to Allain Bertaud, who's got a very interesting work on this stuff: http://alainbertaud.com/

Yes, Fernando, seems like interesting book. I just had a glance on his graphs showing disparities across cities in population density and its dependence on the distance from city center.

Dear Fernando,

I suggest to distinguish between morphological and functional polycentricity. You can, e.g., refer to this paper Meijers and Burger

http://usj.sagepub.com/content/49/5/1127.short

For functional polycentricity indexes, e.g. those involving commuting flows, have a look on work by Nick Green, e.g.:

Green, Nick 2007, 'Functional polycentricity: A formal definition in terms of social network analysis' , Urban Studies, 44 (11) , pp. 2077-2103.

For Morphological polycentricity, often it is referred to works lined to density gradients or to Rank-size rule

I have also wrote something:

http://mpra.ub.uni-muenchen.de/25880/1/MPRA_paper_25880.pdf

http://www.tandfonline.com/doi/abs/10.1080/09654313.2012.673566#.VZaY7PntlHw

There is a paper about urban centrality by Rafael Pereira Moraes et al. at

http://onlinelibrary.wiley.com/doi/10.1111/gean.12002/abstract

Hi Fernando.

What we try to do in the paper mentioned by Bernardo is precisely to propose a simple index to measure the polycentricity of an urban in a continum scale that would go from mono to polycentric pattern.  

It is indeed very important to make the differentiation between functional and morphological polycentricity, as mentioned by David. One of the advantages of the index we propose is that it could be applied both to flow or density data  (e.g. commuting or job distribution). 

area. http://onlinelibrary.wiley.com/doi/10.1111/gean.12002/abstract

 Dear Rafael,

Very interesting abstract! Very expensive download!

Dear Fernando, this paper is freely available in my Research Gate Profile: https://www.researchgate.net/profile/Rafael_Pereira11/publications

If you have any problems downloading the pdf from here, you can try my personal website: https://sites.google.com/site/rafaelhenriquemoraespereira/works-in-progress/publications