On chromatic and flow polynomial unique graphs (original) (raw)

The Tutte polynomial of some matroids

2014

The Tutte polynomial of a graph or a matroid, named after W. T. Tutte, has the important universal property that essentially any mul-tiplicative graph or network invariant with a deletion and contraction reduction must be an evaluation of it. The deletion and contraction operations are natural reductions for many network models arising from a wide range of problems at the heart of computer science, engi-neering, optimization, physics, and biology. Even though the invariant is #P-hard to compute in general, there are many occasions when we face the task of computing the Tutte polynomial for some families of graphs or matroids. In this work we compile known formulas for the Tutte polynomial of some families of graphs and matroids. Also, we give brief explanations of the techniques that were use to find the for-mulas. Hopefully, this will be useful for researchers in Combinatorics and elsewhere.

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...

Basics on Chromatic Polynomials in Graphs

IJCRM.COM, 2016

This article is a general introduction of chromatic polynomials.In this,Chromatic polynomials are defined,their connection between the theory of chromatic polynomials and coloring of graphs.Also it explains the chromatic polynomials of total graphs .It gives the basic concepts of chromatic polynomials in graph theory.

Tutte Polynomials and Graph Symmetries

Symmetry

The Tutte polynomial is an isomorphism invariant of graphs that generalizes the chromatic and the flow polynomials. This two-variable polynomial with integral coefficients is known to carry important information about the properties of the graph. It has been used to prove long-standing conjectures in knot theory. Furthermore, it is related to the Potts and Ising models in statistical physics. The purpose of this paper is to study the interaction between the Tutte polynomial and graph symmetries. More precisely, we prove that if the automorphism group of the graph G contains an element of prime order p, then the coefficients of the Tutte polynomial of G satisfy certain necessary conditions.

Tutte polynomials for directed graphs

Journal of Combinatorial Theory, Series B, 2020

The Tutte polynomial is a fundamental invariant of graphs. In this article, we define and study a generalization of the Tutte polynomial for directed graphs, that we name the B-polynomial. The B-polynomial has three variables, but when specialized to the case of graphs (that is, digraphs where arcs come in pairs with opposite directions), one of the variables becomes redundant and the B-polynomial is equivalent to the Tutte polynomial. We explore various properties, expansions, specializations, and generalizations of the B-polynomial, and try to answer the following questions: • what properties of the digraph can be detected from its B-polynomial (acyclicity, length of directed paths, number of strongly connected components, etc.)? • which of the marvelous properties of the Tutte polynomial carry over to the directed graph setting? The B-polynomial generalizes the strict chromatic polynomial of mixed graphs introduced by Beck, Bogart and Pham. We also consider a quasisymmetric function version of the B-polynomial which simultaneously generalizes the Tutte symmetric function of Stanley and the quasisymmetric chromatic function of Shareshian and Wachs.

Weighted Tutte-Grothendieck polynomials of graphs

arXiv (Cornell University), 2022

In this paper, we introduce the notion of weighted chromatic polynomials of a graph associated to a weight function f of a certain degree, and discuss some of its properties. As a generalization of this concept, we define the weighted Tutte-Grothendieck polynomials of graphs. When f is harmonic, we notice that there is a correspondence between the weighted Tutte-Grothendieck polynomials of graphs and the weighted Tutte polynomials of matroids. Moreover, we present some constructions of the weighted Tutte-Grothendieck invariants for graphs as well as the weighted Tutte invariants for matroids. Finally, we give a remark on the categorification of the weighted chromatic polynomials of graphs and weighted Tutte polynomials of matroids.

Polynomials in graph theory

2015

In this paper we observe the problem of counting graph colorings using polynomials. Several reformulations of The Four Color Conjecture are considered (among them algebraic, probabilistic and arithmetic). In the last section Tutte polynomials are mentioned. 1.

Tutte polynomials for oriented matroids

arXiv (Cornell University), 2022

The Tutte polynomial is a fundamental invariant of graphs and matroids. In this article, we define a generalization of the Tutte polynomial to oriented graphs and regular oriented matroids. To any regular oriented matroid N , we associate a polynomial invariant AN (q, y, z), which we call the A-polynomial. The A-polynomial has the following interesting properties among many others: • a specialization of AN gives the Tutte polynomial of the underlying unoriented matroid N , • when the oriented matroid N corresponds to an unoriented matroid (that is, when the elements of the ground set come in pairs with opposite orientations), the invariant AN is equivalent to the Tutte polynomial of this unoriented matroid (up to a change of variables), • the invariant AN detects, among other things, whether N is acyclic and whether N is totally cyclic. We explore various properties and specializations of the A-polynomial. We show that some of the known properties of the Tutte polynomial of matroids can be extended to the A-polynomial of regular oriented matroids. For instance, we show that a specialization of AN counts all the acyclic orientations obtained by reorienting some elements of N , according to the number of reoriented elements. Let us mention that in a previous article we had defined an invariant of oriented graphs that we called the B-polynomial, which is also a generalization of the Tutte polynomial. However, the B-polynomial of an oriented graph N is not equivalent to its A-polynomial, and the B-polynomial cannot be extended to an invariant of regular oriented matroids.

Some Polynomials of Flower Graphs

2007

We define a class of graphs called flower and give some properties of these graphs. Then the explicit expressions of the chromatic polynomial and the flow polynomial is given. Further, we give classes of graphs with the same chromatic and flow polynomials.