A Polynomial Invariant for Rank 3 Weakly-Colored Stranded Graphs (original) (raw)

Edge colourings and topological graph polynomials

2018

A k-valuation is a special type of edge k-colouring of a medial graph. Various graph polynomials, such as the Tutte, Penrose, Bollobas-Riordan, and transition polynomials, admit combinatorial interpretations and evaluations as weighted counts of k-valuations. In this paper, we consider a multivariate generating function of k-valuations. We show that this is a polynomial in k and hence defines a graph polynomial. We then show that the resulting polynomial has several desirable properties, including a recursive deletion-contraction-type definition, and specialises to the graph polynomials mentioned above. It also offers an alternative extension of the Penrose polynomial from plane graphs to graphs in other surfaces.

Recipe theorems for polynomial invariants on ribbon graphs with half-edges

We provide recipe theorems for the Bollob\`as and Riordan polynomial mathcalR\mathcal{R}mathcalR defined on classes of ribbon graphs with half-edges introduced in arXiv:1310.3708[math.GT]. We also define a generalized transition polynomial QQQ on this new category of ribbon graphs and establish a relationship between QQQ and mathcalR\mathcal{R}mathcalR.

Universality for polynomial invariants on ribbon graphs with flags

In this paper, we analyze the Bollobas and Riordan polynomial for ribbon graphs with flags introduced in arXiv:1301.1987 and prove its universality. We also show that this polynomial can be defined on some equivalence classes of ribbon graphs involving flag moves and that the new polynomial is still universal on these classes.

On terminal forms for topological polynomials for ribbon graphs: The -petal flower

European Journal of Combinatorics, 2014

The Bollobas-Riordan polynomial [Math. Ann. 323, 81 (2002)] extends the Tutte polynomial and its contraction/deletion rule for ordinary graphs to ribbon graphs. Given a ribbon graph G, the related polynomial should be computable from the knowledge of the terminal forms of G namely specific induced graphs for which the contraction/deletion procedure becomes more involved. We consider some classes of terminal forms as rosette ribbon graphs with N ≥ 1 petals and solve their associate Bollobas-Riordan polynomial. This work therefore enlarges the list of terminal forms for ribbon graphs for which the Bollobas-Riordan polynomial could be directly deduced.

Graph Polynomial for Colored Embedded Graphs: A Topological Approach

arXiv (Cornell University), 2022

We study finite graphs embedded in oriented surfaces by associating a polynomial to it. The tools used in developing a theory of such graph polynomials are algebraic topological while the polynomial itself is inspired from ideas arising in physics. We also analyze a variant of these polynomials for colored embedded graphs. This is used to describe the change in the polynomial under basic graph theoretic operations. We conclude with several applications of this polynomial including detection of certain classes of graphs and the connection of this polynomial with topological entanglement entropy.

Edge colouring models for the Tutte polynomial and related graph invariants

For integer q>1, we derive edge q-colouring models for (i) the Tutte polynomial of a graph G on the hyperbola H_q, (ii) the symmetric weight enumerator of the set of group-valued q-flows of G, and (iii) a more general vertex colouring model partition function that includes these polynomials and the principal specialization order q of Stanley's symmetric monochrome polynomial. In the second half of the paper we exhibit a family of non-symmetric edge q-colouring models defined on k-regular graphs, whose partition functions for q >= k each evaluate the number of proper edge k-colourings of G when G is Pfaffian.

Some Specializations and Evaluations of the Tutte Polynomial of a Family of Graphs

Asian journal of natural and applied sciences, 2013

In this paper, we give some specializations and evaluations of the Tutte polynomial of a family of positive-signed connected planar graphs. First of all, we give the general form of the Tutte polynomial of the family of graphs using directly the deletion-contraction definition of the Tutte polynomial. Then, we give general formulas of Jones polynomials of very interesting families of alternating knots and links that correspond to these planar graphs; we actually specialize the Tutte polynomial to the Jones polynomial with the change of variables, and and with some factor of . In case of twocomponent links, we get two different formulas of the Jones polynomial, one when both the links are oriented either in clockwise or counterclockwise direction and another one when one component is oriented clockwise and the second counterclockwise. Moreover, we give general forms of the flow, reliability, and chromatic polynomials of these graphs. The reason to study flow polynomial is that it giv...

The chromatic polynomial of fatgraphs and its categorification

Advances in Mathematics, 2008

Motivated by Khovanov homology and relations between the Jones polynomial and graph polynomials, we construct a homology theory for embedded graphs from which the chromatic polynomial can be recovered as the Euler characteristic. For plane graphs, we show that our chromatic homology can be recovered from the Khovanov homology of an associated link. We apply this connection with Khovanov homology to show that the torsion-free part of our chromatic homology is independent of the choice of planar embedding of a graph.