Locally grid graphs: classification and Tutte uniqueness

We define a locally grid graph as a graph in which the structure around each vertex is a 3×3 grid ⊞, the canonical examples being the toroidal grids Cp×Cq. The paper contains two main results. First, we give a complete classification of locally grid graphs, showing that each of them has a natural embedding in the torus or in the Klein bottle. Secondly, as a continuation of the research initiated in (On graphs determined by their Tutte polynomials, Graphs Combin., to appear), we prove that Cp×Cq is uniquely determined by its Tutte polynomial, for p,q⩾6

On Tutte polynomial uniqueness of twisted wheels

AbstractA graph G is called T-unique if any other graph having the same Tutte polynomial as G is isomorphic to G. Recently, there has been much interest in determining T-unique graphs and matroids. For example, de Mier and Noy [A. de Mier, M. Noy, On graphs determined by their Tutte polynomials, Graphs Combin. 20 (2004) 105–119; A. de Mier, M. Noy, Tutte uniqueness of line graphs, Discrete Math. 301 (2005) 57–65] showed that wheels, ladders, Möbius ladders, square of cycles, hypercubes, and certain class of line graphs are all T-unique. In this paper, we prove that the twisted wheels are also T-unique

Graphs with few matching roots

We determine all graphs whose matching polynomials have at most five distinct zeros. As a consequence, we find new families of graphs which are determined by their matching polynomial.Comment: 14 pages, 7 figures, 1 appendix table. Final version. Some typos are fixe

Hexagonal Tilings: Tutte Uniqueness

We develop the necessary machinery in order to prove that hexagonal tilings are uniquely determined by their Tutte polynomial, showing as an example how to apply this technique to the toroidal hexagonal tiling.Comment: 12 figure

Beta Invariant and Chromatic Uniqueness of Wheels

A graph G is chromatically unique if its chromatic polynomial completely determines the graph. An n-spoked wheel, Wn, is shown to be chromatically unique when n ≥ 4 is even [S.-J. Xu and N.-Z. Li, The chromaticity of wheels, Discrete Math. 51 (1984) 207–212]. When n is odd, this problem is still open for n ≥ 15 since 1984, although it was shown by di erent researchers that the answer is no for n = 5, 7, yes for n = 3, 9, 11, 13, and unknown for other odd n. We use the beta invariant of matroids to prove that if M is a 3-connected matroid such that |E(M)| = |E(Wn)| and β (M) = β (M(Wn)), where β (M) is the beta invariant of M, then M ≅ M(Wn). As a consequence, if G is a 3-connected graph such that the chromatic (or flow) polynomial of G equals to the chromatic (or flow) polynomial of a wheel, then G is isomorphic to the wheel. The examples for n = 3, 5 show that the 3-connectedness condition may not be dropped. We also give a splitting formula for computing the beta invariants of general parallel connection of two matroids as well as the 3-sum of two binary matroids. This generalizes the corresponding result of Brylawski [A combinatorial model for series-parallel networks, Trans. Amer. Math. Soc. 154 (1971) 1–22]

Graphs determined by polynomial invariants

AbstractMany polynomials have been defined associated to graphs, like the characteristic, matchings, chromatic and Tutte polynomials. Besides their intrinsic interest, they encode useful combinatorial information about the given graph. It is natural then to ask to what extent any of these polynomials determines a graph and, in particular, whether one can find graphs that can be uniquely determined by a given polynomial. In this paper we survey known results in this area and, at the same time, we present some new results

Homomorphisms and polynomial invariants of graphs

This paper initiates a general study of the connection between graph homomorphisms and the Tutte polynomial. This connection can be extended to other polynomial invariants of graphs related to the Tutte polynomial such as the transition, the circuit partition, the boundary, and the coboundary polynomials. As an application, we describe in terms of homomorphism counting some fundamental evaluations of the Tutte polynomial in abelian groups and statistical physics. We conclude the paper by providing a homomorphism view of the uniqueness conjectures formulated by Bollobás, Pebody and Riordan.Ministerio de Educación y Ciencia MTM2005-08441-C02-01Junta de Andalucía PAI-FQM-0164Junta de Andalucía P06-FQM-0164

The computation of k-defect polynomials, suspended Y -trees and its applications

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in partial fulfilment of requirements for the degree of Master of Science. June 2014.We start by defining a class of graphs called the suspended Y -trees and give some of its properties. We then classify all the closed sets of a general suspended Y -tree. This will lead us to counting the graph compositions of the suspended Y -tree. We then contract these closed sets one by one to obtain a set of minors for the suspended Y -trees. We will use this information to compute some of the general expression of the k-defect polynomial of a suspended Y -tree. Finally we compute the explicit Tutte polynomial of the suspended Y -trees