Universality for polynomial invariants on ribbon graphs with flags (original) (raw)
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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.
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.
A Polynomial Invariant for Rank 3 Weakly-Colored Stranded Graphs
The Bollobas-Riordan polynomial [Math. Ann. 323, 81 (2002)] is a universal polynomial invariant for ribbon graphs. We find an extension of this polynomial for a particular family of graphs called rank 3 weakly-colored stranded graphs. These graphs live in a 3D space and appear as the gluing stranded vertices with stranded edges according to a definite rule (ordinary graphs and ribbon graphs can be understood in terms of stranded graphs as well). They also possess a color structure in a specific sense [Gurau, Commun. Math. Phys. 304, 69 (2011)]. The polynomial constructed is a seven indeterminate polynomial invariant of these graphs which responds to a similar contraction/deletion recurrence relation as obeyed by the Tutte and Bollobas-Riordan polynomials. It is however new due to the particular cellular structure of the graphs on which it relies. The present polynomial encodes therefore additional data that neither the Tutte nor the Bollobas-Riordan polynomials can capture for the t...
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Let G be a graph and let mij(G), i, j ≥ 1, represents the number of edge of G, where i and j are the degrees of vertices u and v respectively. In this article, we will compute different polynomials of flower graph f(n×m), namely M polynomial and Forgotten polynomial. These polynomials will help us to find many degree based topological indices, included different Zagreb indices, harmonic indices and forgotten index.
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...
2011
We introduce an additional structure on ribbon graphs, arrow structure. We extend the Bollob\'as-Riordan polynomial to ribbon graph with this structure. The extended polynomial satisfies the contraction-deletion relations and naturally behaves with respect to the partial duality of ribbon graphs. We construct an arrow ribbon graph from a virtual link whose extended Bollob\'as-Riordan polynomial specializes to the arrow polynomial of the virtual link recently introduced by H.Dye and L.Kauffman. This result generalizes the classical Thistlethwaite theorem to the arrow polynomial of virtual links.
Possible quantum algorithms for the Bollobas-Riordan-Tutte polynomial of a ribbon graph
Quantum Information and Computation VI, 2008
Three possible quantum algorithms, for the computation of the Bollobás-Riordan-Tutte polynomial of a given ribbon graph, are presented and discussed. The first possible algorithm is based on the spanning quasi-trees expansion for generalized Tutte polynomials of generalized graphs and on a quantum version of the Binary Decision Diagram (BDD) for quasi-trees . The second possible algorithm is based on the relation between the Kauffman bracket and the Tutte polynomial; and with an application of the recently introduced Aharonov-Arad-Eban-Landau quantum algorithm. The third possible algorithm is based on the relation between the HOMFLY polynomial and the Tutte polynomial; and with an application of the Wocjan-Yard quantum algorithm. It is claimed that these possible algorithms may be more efficient that the best known classical algorithms. These three algorithms may have interesting applications in computer science at general or in computational biology and bio-informatics in particular. A line for future research based on the categorification project is mentioned.
Bollobas-Riordan and relative Tutte polynomials
2010
We establish a relation between the Bollobas-Riordan polynomial of a ribbon graph with the relative Tutte polynomial of a plane graph obtained from the ribbon graph using its projection to the plane in a nontrivial way. Also we give a duality formula for the relative Tutte polynomial of dual plane graphs and an expression of the Kauffman bracket of a virtual link as a specialization of the relative Tutte polynomial.