2,723 research outputs found
The Number of Nowhere-Zero Flows on Graphs and Signed Graphs
A nowhere-zero -flow on a graph is a mapping from the edges of
to the set \{\pm1, \pm2, ..., \pm(k-1)\} \subset \bbZ such that, in
any fixed orientation of , at each node the sum of the labels over the
edges pointing towards the node equals the sum over the edges pointing away
from the node. We show that the existence of an \emph{integral flow polynomial}
that counts nowhere-zero -flows on a graph, due to Kochol, is a consequence
of a general theory of inside-out polytopes. The same holds for flows on signed
graphs. We develop these theories, as well as the related counting theory of
nowhere-zero flows on a signed graph with values in an abelian group of odd
order. Our results are of two kinds: polynomiality or quasipolynomiality of the
flow counting functions, and reciprocity laws that interpret the evaluations of
the flow polynomials at negative integers in terms of the combinatorics of the
graph.Comment: 17 pages, to appear in J. Combinatorial Th. Ser.
Weyl invariant polynomial and deformation quantization on Kahler manifolds
Given a polynomial P of partial derivatives of the Kahler metric, expressed
as a linear combination of directed multigraphs, we prove a simple criterion in
terms of the coefficients for to be an invariant polynomial, i.e. invariant
under the transformation of coordinates. As applications, we prove an explicit
composition formula for covariant differential operators under a canonical
basis, also known as invariant differential operators in the case of bounded
symmetric domains. We also prove a general explicit formula of star products on
Kahler manifolds.Comment: 17 page
NLS Bifurcations on the bowtie combinatorial graph and the dumbbell metric graph
We consider the bifurcations of standing wave solutions to the nonlinear
Schr\"odinger equation (NLS) posed on a quantum graph consisting of two loops
connected by a single edge, the so-called dumbbell, recently studied by
Marzuola and Pelinovsky. The authors of that study found the ground state
undergoes two bifurcations, first a symmetry-breaking, and the second which
they call a symmetry-preserving bifurcation. We clarify the type of the
symmetry-preserving bifurcation, showing it to be transcritical. We then reduce
the question, and show that the phenomena described in that paper can be
reproduced in a simple discrete self-trapping equation on a combinatorial graph
of bowtie shape. This allows for complete analysis both by geometric methods
and by parameterizing the full solution space. We then expand the question, and
describe the bifurcations of all the standing waves of this system, which can
be classified into three families, and of which there exists a countably
infinite set
Chromatic Quasisymmetric Class Functions for combinatorial Hopf monoids
We study the chromatic quasisymmetric class function of a linearized combinatorial Hopf monoid. Given a linearized combinatorial Hopf monoid H, and an H-structure h on a set N, there are proper colorings of h, generalizing graph colorings and poset partitions. We show that the automorphism group of h acts on the set of proper colorings. The chromatic quasisymmetric class function enumerates the fixed points of this action, weighting each coloring with a monomial. For the Hopf monoid of graphs this invariant generalizes Stanley\u27s chromatic symmetric function and specializes to the orbital chromatic polynomial of Cameron and Kayibi.
We also introduce the flag quasisymmetric class function of a balanced relative simplicial complex equipped with a group action. We show that, under certain conditions, the chromatic quasisymmetric class function of h is the flag quasisymmetric class function of a balanced relative simplicial complex that we call the coloring complex of h. We use this result to deduce various inequalities for the associated orbital polynomial invariants. We apply these results to several examples related to enumerating graph colorings, poset partitions, generic functions on matroids or generalized permutohedra, and others
Tutte's dichromate for signed graphs
We introduce the ``trivariate Tutte polynomial" of a signed graph as an
invariant of signed graphs up to vertex switching that contains among its
evaluations the number of proper colorings and the number of nowhere-zero
flows. In this, it parallels the Tutte polynomial of a graph, which contains
the chromatic polynomial and flow polynomial as specializations. The number of
nowhere-zero tensions (for signed graphs they are not simply related to proper
colorings as they are for graphs) is given in terms of evaluations of the
trivariate Tutte polynomial at two distinct points. Interestingly, the
bivariate dichromatic polynomial of a biased graph, shown by Zaslavsky to share
many similar properties with the Tutte polynomial of a graph, does not in
general yield the number of nowhere-zero flows of a signed graph. Therefore the
``dichromate" for signed graphs (our trivariate Tutte polynomial) differs from
the dichromatic polynomial (the rank-size generating function).
The trivariate Tutte polynomial of a signed graph can be extended to an
invariant of ordered pairs of matroids on a common ground set -- for a signed
graph, the cycle matroid of its underlying graph and its frame matroid form the
relevant pair of matroids. This invariant is the canonically defined Tutte
polynomial of matroid pairs on a common ground set in the sense of a recent
paper of Krajewski, Moffatt and Tanasa, and was first studied by Welsh and
Kayibi as a four-variable linking polynomial of a matroid pair on a common
ground set.Comment: 53 pp. 9 figure
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