Homomorphisms and Polynomial Invariants of Graphs

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.

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.

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.

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.

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.

Annals of Combinatorics, 2008

We give a new characterization of the Tutte polynomial of graphs. Our characterization is formally close (but inequivalent) to the original definition given by Tutte as the generating function of spanning trees counted according to activities. Tutte's notion of activity requires to choose a linear order on the edge set (though the generating function of the activities is, in fact, independent of this order). We define a new notion of activity, the embedding-activity, which requires to choose a combinatorial embedding of the graph, that is, a cyclic order of the edges around each vertex. We prove that the Tutte polynomial equals the generating function of spanning trees counted according to embedding-activities (this generating function being, in fact, independent of the embedding).

2011

For a basic introduction to graphs and isomorphism see e.g. [35, ch. 4]. For us a graph G = (V, E) may have parallel edges and loops. (Usually called multigraphs or pseudographs. Biggs [4] calls them general graphs, and simple graphs he calls strict graphs.) The sets V and E come with a mapping E → V ∪ ( V 2 ) that assigns to each edge e its endpoint(s). A simple graph has no multiple edges or loops; here an edge e can be identified with an unordered pair of vertices uv (its endpoints u and v).

Symmetry, 2018

In this paper, we study the way the symmetries of a given graph are reflected in its characteristic polynomials. Our aim is not only to find obstructions for graph symmetries in terms of its polynomials but also to measure how faithful these algebraic invariants are with respect to symmetry. Let p be an odd prime and Γ be a finite graph whose automorphism group contains an element h of order p. Assume that the finite cyclic group generated by h acts semi-freely on the set of vertices of Γ with fixed set F. We prove that the characteristic polynomial of Γ, with coefficients in the finite field of p elements, is the product of the characteristic polynomial of the induced subgraph Γ[F] by one of Γ \ F. A similar congruence holds for the characteristic polynomial of the Laplacian matrix of Γ.

Journal of Combinatorial Theory, Series B, 2006

Interpretations for evaluations of the Tutte polynomial T (G; x, y) of a graph G are given at a number of points on the hyperbolae H q = {(x, y) | (x − 1)(y − 1) = q}, for q a positive integer-points at which there are usually no other similarly meaningful graphical interpretations. Further, when q is a prime power, an alternative interpretation for the evaluation of the Tutte polynomial at (1 − q, 0) is presented, more familiarly known as the point which gives the number of proper vertex q-colourings of G.

2019

We construct a new polynomial invariant for signed graphs, the trivariate Tutte polynomial, which contains among its evaluations the number of proper colorings and the number of nowhere-zero flows. In this, it parallels the Tutte polynomial of a graph, which contains the chromatic polynomial and flow polynomial as specializations. While the Tutte polynomial of a graph is equivalently defined as the dichromatic polynomial or Whitney rank polynomial, the dichromatic polynomial of a signed graph (defined more widely for biased graphs by Zaslavsky) does not, by contrast, give the number of nowhere-zero flows as an evaluation in general. The trivariate Tutte polynomial contains Zaslavsky's dichromatic polynomial as a specialization. Furthermore, the trivariate Tutte polynomial gives as an evaluation the number of proper colorings of a signed graph under a more general sense of signed graph coloring in which colors are elements of an arbitrary finite set equipped with an involution.