Tutte sets in graphs II: The complexity of finding maximum Tutte sets

A well-known formula of Tutte and Berge expresses the size of a maximum matching in a graph G in terms of what is usually called the deficiency. A subset X of V(G) for which this deficiency is attained is called a Tutte set of G. While much is known about maximum matchings, less is known about the structure of Tutte sets. We explored the structural aspects of Tutte sets in another paper. Here, we consider the algorithmic complexity of finding Tutte sets in a graph. We first give two polynomial algorithms for finding a maximal Tutte set. We then consider the complexity of finding a maximum Tutte set, and show it is NP-hard for general graphs, as well as for several interesting restricted classes such as planar graphs. By contrast, we show we can find maximum Tutte sets in polynomial time for graphs of level 0 or 1, elementary graphs, and 1-tough graphs.     

On Tutte polynomial uniqueness of twisted wheels

A 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.     

Decomposition of the Flow Polynomial

The flow polynomials denote the number of nowhere-zero flows on graphs, and are related to the well-known Tutte polynomials and chromatic polynomials. We will show the decomposition of the flow polynomials by edge-cuts and vertex-cuts of size 2 or 3. Moreover by using this decomposition, we will consider what kind of graphs have the same flow polynomials. Another application of the decomposition results is that if a bridgeless graph G does not admit a nowhere-zero k-flow and G has a small vertex- or edge-cut, then a proper bridgeless subgraph of G (a graph minor) does not admit a nowhere-zero k-flow either.     

On chromatic and flow polynomial unique graphs

It is known that the chromatic polynomial and flow polynomial of a graph are two important evaluations of its Tutte polynomial, both of which contain much information of the graph. Much research is done on graphs determined entirely by their chromatic polynomials and Tutte polynomials, respectively. Oxley asked which classes of graphs or matroids are determined by their chromatic and flow polynomials together. In this paper, we found several classes of graphs with this property. We first study which graphic parameters are determined by the flow polynomials. Then we study flow-unique graphs. Finally, we show that several classes of graphs, ladders, Möbius ladders and squares of n-cycle are determined by their chromatic polynomials and flow polynomials together. A direct consequence of our theorem is a result of de Mier and Noy [A. de Mier, M. Noy, On graphs determined by their Tutte polynomial, Graphs Comb. 20 (2004) 105-119] that these classes of graphs are Tutte polynomial unique.     

On the colored Tutte polynomial of a graph of bounded treewidth

We observe that a formula given by Negami [Polynomial invariants of graphs, Trans. Amer. Math. Soc. 299 (1987) 601-622] for the Tutte polynomial of a k-sum of two graphs generalizes to a colored Tutte polynomial. Consequently, an algorithm of Andrzejak [An algorithm for the Tutte polynomials of graphs of bounded treewidth, Discrete Math. 190 (1998) 39-54] may be directly adapted to compute the colored Tutte polynomial of a graph of bounded treewidth in polynomial time. This result has also been proven by Makowsky [Colored Tutte polynomials and Kauffman brackets for graphs of bounded tree width, Discrete Appl. Math. 145 (2005) 276-290], using a different algorithm based on logical techniques.     

Orientations, Lattice Polytopes, and Group Arrangements I: Chromatic and Tension Polynomials of Graphs

This is the first one of a series of papers on association of orientations, lattice polytopes, and group arrangements to graphs. The purpose is to interpret the integral and modular tension polynomials of graphs at zero and negative integers. The whole exposition is put under the framework of subgroup arrangements and the application of Ehrhart polynomials. Such a viewpoint leads to the following main results of the paper: (i) the reciprocity law for integral tension polynomials; (ii) the reciprocity law for modular tension polynomials; and (iii) a new interpretation for the value of the Tutte polynomial T(G; x, y) of a graph G at (1, 0) as the number of cut-equivalence classes of acyclic orientations on G.     

On the reconstruction of graph invariants

The reconstruction conjecture has remained open for simple undirected graphs since it was suggested in 1941 by Kelly and Ulam. In an attempt to prove the conjecture, many graph invariants have been shown to be reconstructible from the vertex-deleted deck, and in particular, some prominent graph polynomials. Among these are the Tutte polynomial, the chromatic polynomial and the characteristic polynomial. We show that the interlace polynomial, the U-polynomial, the universal edge elimination polynomial ξ and the colored versions of the latter two are reconstructible.We also present a method of reconstructing boolean graph invariants, or in other words, proving recognizability of graph properties (of colored or uncolored graphs), using first order logic.     

Contraction–Deletion Invariants for Graphs

We consider generalizations of the Tutte polynomial on multigraphs obtained by keeping the main recurrence relation T(G)=T(G/e)+T(G−e) for eE(G) neither a bridge nor a loop and dropping the relations for bridges and loops. Our first aim is to find the universal invariant satisfying these conditions, from which all others may be obtained. Surprisingly, this turns out to be the universal V-function Z of Tutte (1947, Proc. Cambridge Philos. Soc.43, 26–40) defined to obey the same relation for bridges as well. We also obtain a corresponding result for graphs with colours on the edges and describe the universal coloured V-function, which is more complicated than Z. Extending results of Tutte (1974, J. Combin. Theory Ser. B16, 168–174) and Brylawski (1981, J. Combin. Theory Ser. B30, 233–246), we give a simple proof that there are non-isomorphic graphs of arbitrarily high connectivity with the same Tutte polynomial and the same value of Z. We conjecture that almost all graphs are determined by their chromatic or Tutte polynomials and provide mild evidence to support this.     

Graph homomorphisms,the Tutte polynomial and “q-state Potts uniqueness”

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.     

Topological Graph Polynomials in Colored Group Field Theory

In this paper, we analyze the open Feynman graphs of the Colored Group Field Theory introduced in Gurau (Colored group field theory, arXiv:0907.2582 [hep-th]). We define the boundary graph ${\mathcal{G}_{\partial}}In this paper, we analyze the open Feynman graphs of the Colored Group Field Theory introduced in Gurau (Colored group field theory, arXiv:0907.2582 [hep-th]). We define the boundary graph G_?{\mathcal{G}_{\partial}} of an open graph G{\mathcal{G}} and prove it is a cellular complex. Using this structure we generalize the topological (Bollobás–Riordan) Tutte polynomials associated to (ribbon) graphs to topological polynomials adapted to Colored Group Field Theory graphs in arbitrary dimension.     

Tutte sets in graphs I: Maximal tutte sets and D‐graphs

A well‐known formula of Tutte and Berge expresses the size of a maximum matching in a graph G in terms of what is usually called the deficiency of G. A subset X of V(G) for which this deficiency is attained is called a Tutte set of G. While much is known about maximum matchings, less is known about the structure of Tutte sets. In this article, we study the structural aspects of maximal Tutte sets in a graph G. Towards this end, we introduce a related graph D(G). We first show that the maximal Tutte sets in G are precisely the maximal independent sets in its D‐graph D(G), and then continue with the study of D‐graphs in their own right, and of iterated D‐graphs. We show that G is isomorphic to a spanning subgraph of D(G), and characterize the graphs for which G?D(G) and for which D(G)?D²(G). Surprisingly, it turns out that for every graph G with a perfect matching, D³(G)?D²(G). Finally, we characterize bipartite D‐graphs and comment on the problem of characterizing D‐graphs in general. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 343–358, 2007     

Hypergeometric series well-poised in SU(n) and a generalization of Biedenharn's G-functions

We prove that Holman's hypergeometric series well-poised in SU(n) satisfy a general difference equation. We make use of the “path sum” function developed by Biedenharn and this equation to show that a special class of these series, multiplied by simple products, may be regarded as a U(n) generalization of Biedenharn and Louck's G(Δ; X) functions for U(3). The fact that these generalized G-functions are polynomials follows from a detailed study of their symmetries and zeros. As a further application of our general difference equations, we give an elementary proof of Holman's U(n) generalization of the ₅F₄(1) summation theorem.     

Arakawa-Kaneko L-functions and generalized poly-Bernoulli polynomials

We introduce and investigate generalized poly-Bernoulli numbers and polynomials. We state and prove several properties satisfied by these polynomials. The generalized poly-Bernoulli numbers are algebraic numbers. We introduce and study the Arakawa-Kaneko L-functions. The non-positive integer values of the complex variable s of these L-functions are expressed rationally in terms of generalized poly-Bernoulli numbers and polynomials. Furthermore, we prove difference and Raabe?s type formulae for these L-functions.     

Cubic planar hamiltonian graphs of various types

Let U be the set of cubic planar hamiltonian graphs, A the set of graphs G in U such that G-v is hamiltonian for any vertex v of G, B the set of graphs G in U such that G-e is hamiltonian for any edge e of G, and C the set of graphs G in U such that there is a hamiltonian path between any two different vertices of G. With the inclusion and/or exclusion of the sets A,B, and C, U is divided into eight subsets. In this paper, we prove that there is an infinite number of graphs in each of the eight subsets.     

Invariants of Composite Networks Arising as a Tensor Product

We generalize Brylawski’s formula of the Tutte polynomial of a tensor product of matroids to colored connected graphs, matroids, and disconnected graphs. Unlike the non-colored tensor product where all edges have to be replaced by the same graph, our colored generalization of the tensor product operation allows individual edge replacement. The colored Tutte polynomials we compute exists by the results of Bollobás and Riordan. The proof depends on finding the correct generalization of the two components of the pointed Tutte polynomial, first studied by Brylawski and Oxley, and on careful enumeration of the connected components in a tensor product. Our results make the calculation of certain invariants of many composite networks easier, provided that the invariants are obtained from the colored Tutte polynomials via substitution and the composite networks are represented as tensor products of colored graphs. In particular, our method can be used to calculate (with relative ease) the expected number of connected components after an accident hits a composite network in which some major links are identical subnetworks in themselves.      

Fixing subgraphs and Ulam's conjecture

A spanning subgraph U of a graph G belongs to the set $\text{J}$(G) of fixing subgraphs (see [5]) of G if every embedding of U into G can be extended to an automorphism of G. Clearly G ∈ $\text{J}$(G). G is free if …$\text{J}$(G)… = 1. We establish a connection between Ulam's conjecture and free graphs and continue with an investigation of free graphs.     

Zeta functions of line, middle, total graphs of a graph and their coverings

We consider the (Ihara) zeta functions of line graphs, middle graphs and total graphs of a regular graph and their (regular or irregular) covering graphs. Let L(G), M(G) and T(G) denote the line, middle and total graph of G, respectively. We show that the line, middle and total graph of a (regular and irregular, respectively) covering of a graph G is a (regular and irregular, respectively) covering of L(G), M(G) and T(G), respectively. For a regular graph G, we express the zeta functions of the line, middle and total graph of any (regular or irregular) covering of G in terms of the characteristic polynomial of the covering. Also, the complexities of the line, middle and total graph of any (regular or irregular) covering of G are computed. Furthermore, we discuss the L-functions of the line, middle and total graph of a regular graph G.     

Tutte polynomials for counting and classifying orbits

Given a graph Γ and an automorphism group , we define some polynomials which count and classify the orbits of G on various structures on Γ, as counted by the Tutte polynomial, while also specialising to the Tutte polynomial.     

On the roots of independence polynomials of almost all very well-covered graphs

If s_k denotes the number of stable sets of cardinality k in graph G, and α(G) is the size of a maximum stable set, then is the independence polynomial of G [I. Gutman, F. Harary, Generalizations of the matching polynomial, Utilitas Math. 24 (1983) 97-106]. A graph G is very well-covered [O. Favaron, Very well-covered graphs, Discrete Math. 42 (1982) 177-187] if it has no isolated vertices, its order equals 2α(G) and it is well-covered, i.e., all its maximal independent sets are of the same size [M.D. Plummer, Some covering concepts in graphs, J. Combin. Theory 8 (1970) 91-98]. For instance, appending a single pendant edge to each vertex of G yields a very well-covered graph, which we denote by G^*. Under certain conditions, any well-covered graph equals G^* for some G [A. Finbow, B. Hartnell, R.J. Nowakowski, A characterization of well-covered graphs of girth 5 or greater, J. Combin. Theory Ser B 57 (1993) 44-68].The root of the smallest modulus of the independence polynomial of any graph is real [J.I. Brown, K. Dilcher, R.J. Nowakowski, Roots of independence polynomials of well-covered graphs, J. Algebraic Combin. 11 (2000) 197-210]. The location of the roots of the independence polynomial in the complex plane, and the multiplicity of the root of the smallest modulus are investigated in a number of articles.In this paper we establish formulae connecting the coefficients of I(G;x) and I(G^*;x), which allow us to show that the number of roots of I(G;x) is equal to the number of roots of I(G^*;x) different from -1, which appears as a root of multiplicity α(G^*)-α(G) for I(G^*;x). We also prove that the real roots of I(G^*;x) are in [-1,-1/2α(G^*)), while for a general graph of order n we show that its roots lie in |z|>1/(2n-1).Hoede and Li [Clique polynomials and independent set polynomials of graphs, Discrete Math. 125 (1994) 219-228] posed the problem of finding graphs that can be uniquely defined by their clique polynomials (clique-unique graphs). Stevanovic [Clique polynomials of threshold graphs, Univ. Beograd Publ. Elektrotehn. Fac., Ser. Mat. 8 (1997) 84-87] proved that threshold graphs are clique-unique. Here, we demonstrate that the independence polynomial distinguishes well-covered spiders among well-covered trees.     

The smallest values of algebraic connectivity for unicyclic graphs

The algebraic connectivity of G is the second smallest eigenvalue of its Laplacian matrix. Let U_n be the set of all unicyclic graphs of order n. In this paper, we will provide the ordering of unicyclic graphs in U_n up to the last seven graphs according to their algebraic connectivities when n≥13. This extends the results of Liu and Liu [Y. Liu, Y. Liu, The ordering of unicyclic graphs with the smallest algebraic connectivity, Discrete Math. 309 (2009) 4315-4325] and Guo [J.-M. Guo, A conjecture on the algebraic connectivity of connected graphs with fixed girth, Discrete Math. 308 (2008) 5702-5711].