1,524 research outputs found
Algebraic number-theoretic properties of graph and matroid polynomials
PhDThis thesis is an investigation into the algebraic number-theoretical
properties of certain polynomial invariants of graphs and matroids.
The bulk of the work concerns chromatic polynomials of graphs,
and was motivated by two conjectures proposed during a 2008 Newton
Institute workshop on combinatorics and statistical mechanics.
The first of these predicts that, given any algebraic integer, there is
some natural number such that the sum of the two is the zero of a
chromatic polynomial (chromatic root); the second that every positive
integer multiple of a chromatic root is also a chromatic root.
We compute general formulae for the chromatic polynomials of two
large families of graphs, and use these to provide partial proofs of
each of these conjectures. We also investigate certain correspondences
between the abstract structure of graphs and the splitting
fields of their chromatic polynomials.
The final chapter concerns the much more general multivariate
Tutte polynomials—or Potts model partition functions—of matroids.
We give three separate proofs that the Galois group of every
such polynomial is a direct product of symmetric groups, and conjecture
that an analogous result holds for the classical bivariate Tutte
polynomial
On Brenti's conjecture about the log-concavity of the chromatic polynomial
The chromatic polynomial is a well studied object in graph theory. There are
many results and conjectures about the log-concavity of the chromatic
polynomial and other polynomials related to it. The location of the roots of
these polynomials has also been well studied. One famous result due to A. Sokal
and C. Borgs provides a bound on the absolute value of the roots of the
chromatic polynomial in terms of the highest degree of the graph. We use this
result to prove a modification of a log-concavity conjecture due to F. Brenti.
The original conjecture of Brenti was that the chromatic polynomial is
log-concave on the natural numbers. This was disproved by Paul Seymour by
presenting a counter example. We show that the chromatic polynomial of
graph is in fact log-concave for all for an explicit
constant , where denotes the highest degree of . We also
provide an example which shows that the result is not true for constants
smaller than 1
Chromatic roots and limits of dense graphs
In this short note we observe that recent results of Abert and Hubai and of
Csikvari and Frenkel about Benjamini--Schramm continuity of the holomorphic
moments of the roots of the chromatic polynomial extend to the theory of dense
graph sequences. We offer a number of problems and conjectures motivated by
this observation.Comment: 9 page
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