1,838 research outputs found
A Penrose polynomial for embedded graphs
We extend the Penrose polynomial, originally defined only for plane graphs,
to graphs embedded in arbitrary surfaces. Considering this Penrose polynomial
of embedded graphs leads to new identities and relations for the Penrose
polynomial which can not be realized within the class of plane graphs. In
particular, by exploiting connections with the transition polynomial and the
ribbon group action, we find a deletion-contraction-type relation for the
Penrose polynomial. We relate the Penrose polynomial of an orientable
checkerboard colourable graph to the circuit partition polynomial of its medial
graph and use this to find new combinatorial interpretations of the Penrose
polynomial. We also show that the Penrose polynomial of a plane graph G can be
expressed as a sum of chromatic polynomials of twisted duals of G. This allows
us to obtain a new reformulation of the Four Colour Theorem
Evaluations of topological Tutte polynomials
We find new properties of the topological transition polynomial of embedded
graphs, . We use these properties to explain the striking similarities
between certain evaluations of Bollob\'as and Riordan's ribbon graph
polynomial, , and the topological Penrose polynomial, . The general
framework provided by also leads to several other combinatorial
interpretations these polynomials. In particular, we express , ,
and the Tutte polynomial, , as sums of chromatic polynomials of graphs
derived from ; show that these polynomials count -valuations of medial
graphs; show that counts edge 3-colourings; and reformulate the Four
Colour Theorem in terms of . We conclude with a reduction formula for the
transition polynomial of the tensor product of two embedded graphs, showing
that it leads to additional relations among these polynomials and to further
combinatorial interpretations of and .Comment: V2: major revision, several new results, and improved expositio
On the interplay between embedded graphs and delta-matroids
The mutually enriching relationship between graphs and matroids has motivated discoveries
in both fields. In this paper, we exploit the similar relationship between embedded graphs and
delta-matroids. There are well-known connections between geometric duals of plane graphs and
duals of matroids. We obtain analogous connections for various types of duality in the literature
for graphs in surfaces of higher genus and delta-matroids. Using this interplay, we establish a
rough structure theorem for delta-matroids that are twists of matroids, we translate Petrie duality
on ribbon graphs to loop complementation on delta-matroids, and we prove that ribbon graph
polynomials, such as the Penrose polynomial, the characteristic polynomial, and the transition
polynomial, are in fact delta-matroidal. We also express the Penrose polynomial as a sum of
characteristic polynomials
Hopf algebras and Tutte polynomials
By considering Tutte polynomials of Hopf algebras, we show how a Tutte
polynomial can be canonically associated with combinatorial objects that have
some notions of deletion and contraction. We show that several graph
polynomials from the literature arise from this framework. These polynomials
include the classical Tutte polynomial of graphs and matroids, Las Vergnas'
Tutte polynomial of the morphism of matroids and his Tutte polynomial for
embedded graphs, Bollobas and Riordan's ribbon graph polynomial, the Krushkal
polynomial, and the Penrose polynomial.
We show that our Tutte polynomials of Hopf algebras share common properties
with the classical Tutte polynomial, including deletion-contraction
definitions, universality properties, convolution formulas, and duality
relations. New results for graph polynomials from the literature are then
obtained as examples of the general results.
Our results offer a framework for the study of the Tutte polynomial and its
analogues in other settings, offering the means to determine the properties and
connections between a wide class of polynomial invariants.Comment: v2: change of title and some reorderin
Structure of the flow and Yamada polynomials of cubic graphs
We establish a quadratic identity for the Yamada polynomial of ribbon cubic
graphs in 3-space, extending the Tutte golden identity for planar cubic graphs.
An application is given to the structure of the flow polynomial of cubic graphs
at zero. The golden identity for the flow polynomial is conjectured to
characterize planarity of cubic graphs, and we prove this conjecture for a
certain infinite family of non-planar graphs.
Further, we establish exponential growth of the number of chromatic
polynomials of planar triangulations, answering a question of D. Treumann and
E. Zaslow. The structure underlying these results is the chromatic algebra, and
more generally the SO(3) topological quantum field theory.Comment: 22 page
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