This paper is devoted to a combinatorial problem for incidence semirings, which can be viewed as sets of polynomials over graphs, where the edges are the unknowns and the coefficients are taken from a semiring. The construction of incidence rings is very well known and has many useful applications. The present article is devoted to a novel application of the more general incidence semirings. Recent research on data mining has motivated the investigation of the sets of centroids that have largest weights in semiring constructions. These sets are valuable for the design of centroid-based classification systems, or classifiers, as well as for the design of multiple classifiers combining several individual classifiers. Our article gives a complete description of all sets of centroids with the largest weight in incidence semirings.
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