Graph polynomials and symmetries

Linear Algebra and its Applications, 1983

In this paper, we study the characteristic polynomials of graphs which admit semifree actions of an abelian group. Using the method of group matrices, we are able to show that the characteristic polynomial of a such a graph is factorized into a product of a polynomial associated to the orbit of the action and a polynomial associated to the free part of the action.

Combinatorics, Probability and Computing, 2009

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 generalize them which also captures Tutte's universal V -functions as a specialization. We show that the equivalence remains true for the extended functions thus answering a question raised by Dominic Welsh.

TURKISH JOURNAL OF MATHEMATICS, 2015

We give explicit expressions of the Tutte polynomial of asymmetric complete flower graph and asymmetric incomplete flower graph. We then express these Tutte polynomials as generating functions and decode some valuable information about the asymmetric complete flower graph and asymmetric incomplete flower graph. Furthermore, we convert the Tutte polynomials into coboundary polynomials and give explicit expressions of the k-defect polynomials of these structures. Finally, we conclude that nonisomorphic graphs in this class have the same Tutte polynomials, the same chromatic polynomials, and the same defect polynomials.

The Seventh European Conference on Combinatorics, Graph Theory and Applications, 2013

We give a method of generating strongly polynomial sequences of graphs, i.e., sequences (H k) indexed by a multivariate parameter k = (k1,. .. , k h) such that, for each fixed graph G, there is a multivariate polynomial p(G; x1,. .. , x h) such that the number of homomorphisms from G to H k is given by the evaluation p(G; k1,. .. , k h). Our construction produces a large family of graph polynomials that includes the Tutte polynomial, the Averbouch-Godlin-Makowsky polynomial and the Tittmann-Averbouch-Makowsky polynomial. We also introduce a new graph parameter, the branching core size of a simple graph, related to how many involutive automorphisms with fixed points it has. We prove that a countable family of graphs of bounded branching core size (which in particular implies bounded tree-depth) can always be partitioned into a finite number of strongly polynomial subsequences.

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.

Turkish Journal of Mathematics

We give explicit expressions of the Tutte polynomial of asymmetric complete flower graph and asymmetric incomplete flower graph. We then express these Tutte polynomials as generating functions and decode some valuable information about the asymmetric complete flower graph and asymmetric incomplete flower graph. Furthermore, we convert the Tutte polynomials into coboundary polynomials and give explicit expressions of the k-defect polynomials of these structures. Finally, we conclude that nonisomorphic graphs in this class have the same Tutte polynomials, the same chromatic polynomials, and the same defect polynomials.

Symmetry

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.

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.

2008

This paper is based on a series of talks given at the Patejdlovka Enumeration Workshop held in the Czech Republic in November 2007. The topics covered are as follows. The graph polynomial, Tutte-Grothendieck invariants, an overview of relevant elementary finite Fourier analysis, the Tutte polynomial of a graph as a Hamming weight enumerator of its set of tensions (or flows), description of a family of polynomials containing the graph polynomial which yield Tutte-Grothendieck invariants in a similar way.

Electronic Notes in Discrete Mathematics, 2009

We establish for which weighted graphs H homomorphism functions from multigraphs G to H are specializations of the Tutte polynomial of G, answering a question of Freedman, Lovász and Schrijver. We introduce a new property of graphs called "q-state Potts uniqueness" and relate it to chromatic and Tutte uniqueness, and also to "chromatic-flow uniqueness", recently studied by Duan, Wu and Yu.