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I am working across mathematics, physics and engineering. And I am looking for whether there exists already formally established knowledge in the field.

Given a periodic graph (actually a physical lattice or crystal structure), we want to examine some periodic coloring ( or ordered but not periodic coloring) of the graph. This coloring in physics may be considered as filling the lattice site with atoms, ions and so forth. Is there some already formally established mathematical area that deals with the this?

Further, we could construct functions from these kinds of coloring to a real number. ( in physics we associate average energy with the specific structure) and we want to minimize it by searching the periodic or ordered graph space. Is there such a branch in mathematics dealing with this also?

Furthermore, I wish to construct a program to analyze any arbitrary periodic colored graphs (that is to analyze different crystal structure. ) By analyze I mean search the coloring that would give the minimum energy. Is there existing well established knowledge for doing this?

An reduced problem would be is there some established mathematical theorems tackling the coloring of an arbitrary periodic graph, in any possible way?

Thank you very much.

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There is a theory of color symmetry in mathematical crystallography and in particular, it is applied to lattices. The basic idea is that each coset of a sublattice is assigned a unique color, and what one usually aims for is a symmetrically colored lattice.

Some introductory papers on the topics are: 1.Color groups associated with square and hexagonal lattices 2. Bravais colourings of planar modules with N-fold symmetry

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In the Mathematics of long range order there is some interest in multi-colored sets. Most of the studied problems are related to diffraction of such models, I think the problem you ask is related to the entropy of such systems.

This paper might or might not be of interest to you:

"Consequences of Pure Point Diffraction Spectra for Multiset Substitution Systems."

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I am not quite sure how to interpret your first question. But you may try taking a look at an old book by Norman Biggs called Interaction Models (Cambridge Univ. Press 1976).

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