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.

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.

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.

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 Absolutely split metacyclic groups and weak metacirculants

2019 ◽  
Vol 18 (06) ◽  
pp. 1950117 ◽  
Author(s):  
Li Cui ◽  
Jin-Xin Zhou
Keyword(s):  
Automorphism Group ◽  
Prime Power ◽  
Necessary Condition ◽  
Prime Power Order ◽  
Metacyclic Groups ◽  
Vertex Transitive ◽  
Open Question ◽  
Positive Integers ◽  
The Relationship ◽  
Sufficient And Necessary Condition

Let [Formula: see text] be positive integers, and let [Formula: see text] be a split metacyclic group such that [Formula: see text]. We say that [Formula: see text] is absolutely split with respect to[Formula: see text] provided that for any [Formula: see text], if [Formula: see text], then there exists [Formula: see text] such that [Formula: see text] and [Formula: see text]. In this paper, we give a sufficient and necessary condition for the group [Formula: see text] being absolutely split. This generalizes a result of Sanming Zhou and the second author in [Weak metacirculants of odd prime power order, J. Comb. Theory A 155 (2018) 225–243]. We also use this result to investigate the relationship between metacirculants and weak metacirculants. Metacirculants were introduced by Alspach and Parsons in [Formula: see text] and have been a rich source of various topics since then. As a generalization of this class of graphs, Marušič and Šparl in 2008 introduced the so-called weak metacirculants. A graph is called a weak metacirculant if it has a vertex-transitive metacyclic automorphism group. In this paper, it is proved that a weak metacirculant of [Formula: see text]-power order is a metacirculant if and only if it has a vertex-transitive split metacyclic automorphism group. This provides a partial answer to an open question in the literature.

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 SOME SPECIAL FAMILIES OF HYPERELLIPTIC CURVES

2004 ◽  
Vol 03 (01) ◽  
pp. 75-89 ◽  
Author(s):  
TANUSH SHASKA
Keyword(s):  
Automorphism Group ◽  
Function Field ◽  
Hyperelliptic Curves ◽  
Necessary Condition ◽  
Field Of Moduli ◽  
Field Of Definition ◽  
G 12

Let [Formula: see text] denote the locus of hyperelliptic curves of genus g whose automorphism group contains a subgroup isomorphic to G. We study spaces [Formula: see text] for G≅ℤn, ℤ2⊕ℤn, ℤ2⊕A4, or SL2(3). We show that for G≅ℤn, ℤ2⊕ℤn, the space [Formula: see text] is a rational variety and find generators of its function field. For G≅ℤ2⊕A4, SL2(3) we find a necessary condition in terms of the coefficients, whether or not the curve belongs to [Formula: see text]. Further, we describe algebraically the loci of such curves for g≤12 and show that for all curves in these loci, the field of moduli is a field of definition.

scholarly journals On the Degeneracy of the Orbit Polynomial and Related Graph Polynomials

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1643
Author(s):  
Modjtaba Ghorbani ◽  
Matthias Dehmer ◽  
Frank Emmert-Streib
Keyword(s):  
Automorphism Group ◽  
Structural Properties ◽  
Graph Polynomials ◽  
Graph Polynomial ◽  
Related Graph ◽  
Graph Operation ◽  
Counting Polynomial

The orbit polynomial is a new graph counting polynomial which is defined as OG(x)=∑i=1rx|Oi|, where O1, …, Or are all vertex orbits of the graph G. In this article, we investigate the structural properties of the automorphism group of a graph by using several novel counting polynomials. Besides, we explore the orbit polynomial of a graph operation. Indeed, we compare the degeneracy of the orbit polynomial with a new graph polynomial based on both eigenvalues of a graph and the size of orbits.

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.

The Infancy of Solar-Type Stars and its Relevance for the Origin of Life

1997 ◽  
Vol 161 ◽  
pp. 267-282 ◽  
Author(s):  
Thierry Montmerle
Keyword(s):  
Main Sequence ◽  
Solar Nebula ◽  
Circumstellar Disks ◽  
T Tauri Stars ◽  
Necessary Condition ◽  
Sequence Evolution ◽  
T Tauri ◽  
Solar Type ◽  
Optical Domain ◽  
The Origin Of Life

AbstractFor life to develop, planets are a necessary condition. Likewise, for planets to form, stars must be surrounded by circumstellar disks, at least some time during their pre-main sequence evolution. Much progress has been made recently in the study of young solar-like stars. In the optical domain, these stars are known as «T Tauri stars». A significant number show IR excess, and other phenomena indirectly suggesting the presence of circumstellar disks. The current wisdom is that there is an evolutionary sequence from protostars to T Tauri stars. This sequence is characterized by the initial presence of disks, with lifetimes ~ 1-10 Myr after the intial collapse of a dense envelope having given birth to a star. While they are present, about 30% of the disks have masses larger than the minimum solar nebula. Their disappearance may correspond to the growth of dust grains, followed by planetesimal and planet formation, but this is not yet demonstrated.

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