352 research outputs found
An elementary chromatic reduction for gain graphs and special hyperplane arrangements
A gain graph is a graph whose edges are labelled invertibly by "gains" from a
group. "Switching" is a transformation of gain graphs that generalizes
conjugation in a group. A "weak chromatic function" of gain graphs with gains
in a fixed group satisfies three laws: deletion-contraction for links with
neutral gain, invariance under switching, and nullity on graphs with a neutral
loop. The laws lead to the "weak chromatic group" of gain graphs, which is the
universal domain for weak chromatic functions. We find expressions, valid in
that group, for a gain graph in terms of minors without neutral-gain edges, or
with added complete neutral-gain subgraphs, that generalize the expression of
an ordinary chromatic polynomial in terms of monomials or falling factorials.
These expressions imply relations for chromatic functions of gain graphs.
We apply our relations to some special integral gain graphs including those
that correspond to the Shi, Linial, and Catalan arrangements, thereby obtaining
new evaluations of and new ways to calculate the zero-free chromatic polynomial
and the integral and modular chromatic functions of these gain graphs, hence
the characteristic polynomials and hypercubical lattice-point counting
functions of the arrangements. We also calculate the total chromatic polynomial
of any gain graph and especially of the Catalan, Shi, and Linial gain graphs.Comment: 31 page
Coclass theory for nilpotent semigroups via their associated algebras
Coclass theory has been a highly successful approach towards the
investigation and classification of finite nilpotent groups. Here we suggest a
similar approach for finite nilpotent semigroups. This differs from the group
theory setting in that we additionally use certain algebras associated to the
considered semigroups. We propose a series of conjectures on our suggested
approach. If these become theorems, then this would reduce the classification
of nilpotent semigroups of a fixed coclass to a finite calculation. Our
conjectures are supported by the classification of nilpotent semigroups of
coclass 0 and 1. Computational experiments suggest that the conjectures also
hold for the nilpotent semigroups of coclass 2 and 3.Comment: 13 pages, 2 figure
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