Graphs, ribbon graphs, and polynomials

2021 ◽  
pp. 7-16
Author(s):  
Adrian Tanasa
Keyword(s):  
Graph Theory ◽  
Tutte Polynomial ◽  
Graph Polynomials ◽  
Ribbon Graphs

In this chapter we present some notions of graph theory that will be useful in the rest of the book. It is worth emphasizing that graph theorists and theoretical physicists adopt, unfortunately, different terminologies. We present here both terminologies, such that a sort of dictionary between these two communities can be established. We then extend the notion of graph to that of maps (or of ribbon graphs). Moreover, graph polynomials encoding these structures (the Tutte polynomial for graphs and the Bollobás–Riordan polynomial for ribbon graphs) are presented.

Types of embedded graphs and their Tutte polynomials

2019 ◽  
Vol 169 (2) ◽  
pp. 255-297 ◽  
Author(s):  
STEPHEN HUGGETT ◽  
IAIN MOFFATT
Keyword(s):  
Systematic Approach ◽  
Tutte Polynomial ◽  
Embedded Graphs ◽  
Graph Polynomials ◽  
Ribbon Graphs ◽  
Duality Relations ◽  
Graphs On Surfaces ◽  
Tutte Polynomials

AbstractWe take an elementary and systematic approach to the problem of extending the Tutte polynomial to the setting of embedded graphs. Four notions of embedded graphs arise naturally when considering deletion and contraction operations on graphs on surfaces. We give a description of each class in terms of coloured ribbon graphs. We then identify a universal deletion-contraction invariant (i.e., a ‘Tutte polynomial’) for each class. We relate these to graph polynomials in the literature, including the Bollobás–Riordan, Krushkal and Las Vergnas polynomials, and give state-sum formulations, duality relations, deleton-contraction relations, and quasi-tree expansions for each of them.

scholarly journals Graph polynomials and symmetries

2019 ◽  
Vol 18 (09) ◽  
pp. 1950172 ◽  
Author(s):  
Nafaa Chbili
Keyword(s):  
Automorphism Group ◽  
Finite Field ◽  
Characteristic Polynomial ◽  
Prime Order ◽  
Tutte Polynomial ◽  
Quotient Graph ◽  
Graph Polynomials ◽  
Tutte Polynomials

In a recent paper, we studied the interaction between the automorphism group of a graph and its Tutte polynomial. More precisely, we proved that certain symmetries of graphs are clearly reflected by their Tutte polynomials. The purpose of this paper is to extend this study to other graph polynomials. In particular, we prove that if a graph [Formula: see text] has a symmetry of prime order [Formula: see text], then its characteristic polynomial, with coefficients in the finite field [Formula: see text], is determined by the characteristic polynomial of its quotient graph [Formula: see text]. Similar results are also proved for some generalization of the Tutte polynomial.

scholarly journals Graphs, Links, and Duality on Surfaces

2010 ◽  
Vol 20 (2) ◽  
pp. 267-287 ◽  
Author(s):  
VYACHESLAV KRUSHKAL
Keyword(s):  
Planar Graphs ◽  
Jones Polynomial ◽  
Tutte Polynomial ◽  
Natural Duality ◽  
Polynomial Invariant ◽  
Topological Duality ◽  
Ribbon Graphs ◽  
Virtual Links ◽  
Graphs On Surfaces ◽  
Tutte Polynomials

We introduce a polynomial invariant of graphs on surfaces,PG, generalizing the classical Tutte polynomial. Topological duality on surfaces gives rise to a natural duality result forPG, analogous to the duality for the Tutte polynomial of planar graphs. This property is important from the perspective of statistical mechanics, where the Tutte polynomial is known as the partition function of the Potts model. For ribbon graphs,PGspecializes to the well-known Bollobás–Riordan polynomial, and in fact the two polynomials carry equivalent information in this context. Duality is also established for a multivariate version of the polynomialPG. We then consider a 2-variable version of the Jones polynomial for links in thickened surfaces, taking into account homological information on the surface. An analogue of Thistlethwaite's theorem is established for these generalized Jones and Tutte polynomials for virtual links.

scholarly journals Graph polynomials and link invariants as positive type functions on Thompson’s group F

2019 ◽  
Vol 28 (02) ◽  
pp. 1950006 ◽  
Author(s):  
Valeriano Aiello ◽  
Roberto Conti
Keyword(s):  
Elementary Proof ◽  
Tutte Polynomial ◽  
Positive Type ◽  
Polynomial Invariants ◽  
Graph Polynomials ◽  
Link Invariants ◽  
Kauffman Bracket ◽  
Model Interpretation ◽  
Mechanics Model ◽  
Thompson’S Group

In a recent paper, Jones introduced a correspondence between elements of the Thompson group [Formula: see text] and certain graphs/links. It follows from his work that several polynomial invariants of links, such as the Kauffman bracket, can be reinterpreted as coefficients of certain unitary representations of [Formula: see text]. We give a somewhat different and elementary proof of this fact for the Kauffman bracket evaluated at certain roots of unity by means of a statistical mechanics model interpretation. Moreover, by similar methods we show that, for some particular specializations of the variables, other familiar link invariants and graph polynomials, namely the number of [Formula: see text]-colorings and the Tutte polynomial, can be viewed as positive definite functions on [Formula: see text].

scholarly journals A Polynomial Invariant and Duality for Triangulations

10.37236/4162 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
Vyacheslav Krushkal ◽  
David Renardy
Keyword(s):  
Planar Graphs ◽  
Essential Feature ◽  
Dual Graph ◽  
Tutte Polynomial ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
Ribbon Graphs ◽  
Higher Dimensional ◽  
Classical Invariant ◽  
Dimension 2

The Tutte polynomial ${T}_G(X,Y)$ of a graph $G$ is a classical invariant, important in combinatorics and statistical mechanics. An essential feature of the Tutte polynomial is the duality for planar graphs $G$, $T_G(X,Y) = {T}_{G^*}(Y,X)$ where $G^*$ denotes the dual graph. We examine this property from the perspective of manifold topology, formulating polynomial invariants for higher-dimensional simplicial complexes. Polynomial duality for triangulations of a sphere follows as a consequence of Alexander duality. The main goal of this paper is to introduce and begin the study of a more general $4$-variable polynomial for triangulations and handle decompositions of orientable manifolds. Polynomial duality in this case is a consequence of Poincaré duality on manifolds. In dimension 2 these invariants specialize to the well-known polynomial invariants of ribbon graphs defined by B. Bollobás and O. Riordan. Examples and specific evaluations of the polynomials are discussed.

scholarly journals The Equivalence of Two Graph Polynomials and a Symmetric Function

2009 ◽  
Vol 18 (4) ◽  
pp. 601-615 ◽  
Author(s):  
CRIEL MERINO ◽  
STEVEN D. NOBLE
Keyword(s):  
Symmetric Function ◽  
Tutte Polynomial ◽  
Graph Polynomials ◽  
Definition Of ◽  
Natural Way

The U-polynomial, the polychromate and the symmetric function generalization of the Tutte polynomial, due to Stanley, are known to be equivalent in the sense that the coefficients of any one of them can be obtained as a function of the coefficients of any other. The definition of each of these functions suggests a natural way in which to strengthen them, which also captures Tutte's universal V-function as a specialization. We show that the equivalence remains true for the strong functions, thus answering a question raised by Dominic Welsh.

scholarly journals Graph Polynomials

ISRN Algebra ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-10
Author(s):  
Mehdi Alaeiyan ◽  
Saeid Mohammadian
Keyword(s):  
Graph Theory ◽  
Linear Equations ◽  
Vertex Cover ◽  
Dominating Sets ◽  
System Of Linear Equations ◽  
Binary Programming ◽  
Edge Cover ◽  
Graph Polynomials

One of the most important and applied concepts in graph theory is to find the edge cover, vertex cover, and dominating sets with minimum cardinal also to find independence and matching sets with maximum cardinal and their polynomials. Although there exist some algorithms for finding some of them (Kuhn and Wattenhofer, 2003; and Mihelic and Robic, 2005), but in this paper we want to study all of these concepts from viewpoint linear and binary programming and we compute the coefficients of the polynomials by solving a system of linear equations with variables.

A NOTE ON THE TUTTE POLYNOMIAL AND THE AUTOMORPHISM GROUP OF A GRAPH

2014 ◽  
Vol 07 (01) ◽  
pp. 1450001 ◽  
Author(s):  
Nafaa Chbili
Keyword(s):  
Automorphism Group ◽  
Short Note ◽  
Tutte Polynomial ◽  
Necessary Condition ◽  
Graph Polynomials

A graph G is said to be p-periodic if the automorphism group Aut(G) contains an element of order p which preserves no edges. In this short note, we investigate the behavior of graph polynomials (Negami and Tutte) with respect to graph periodicity. In particular, we prove that if p is a prime, then the coefficients of the Tutte polynomial of such a graph satisfy a certain necessary condition.

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