Downloads & Free Reading Options - Results

We consider the asymptotic normalcy of families of random variables $X$ which count the number of occupied sites in some large set. We write $Prob(X=m)=p_mz_0^m/P(z_0)$, where $P(z)$ is the generating function $P(z)=\sum_{j=0}^{N}p_jz^j$ and $z_0>0$. We give sufficient criteria, involving the location of the zeros of $P(z)$, for these families to satisfy a central limit theorem (CLT) and even a local CLT (LCLT); the theorems hold in the sense of estimates valid for large $N$ (we assume that $Var(X)$ is large when $N$ is). For example, if all the zeros lie in the closed left half plane then $X$ is asymptotically normal, and when the zeros satisfy some additional conditions then $X$ satisfies an LCLT. We apply these results to cases in which $X$ counts the number of edges in the (random) set of "occupied" edges in a graph, with constraints on the number of occupied edges attached to a given vertex. Our results also apply to systems of interacting particles, with $X$ counting the number of particles in a box $\Lambda$ whose size approaches infinity; $P(z)$ is then the grand canonical partition function and its zeros are the Lee-Yang zeros.

We consider a family of I-graphs I(n,k,l), which is a generalization of the class of generalized Petersen graphs. In the present paper, we provide a new method for counting Jacobian group of the I-graph I(n,k,l). We show that the minimum number of generators of Jac(I(n,k,l)) is at least two and at most 2k + 2l - 1. Also, we obtain a closed formula for the number of spanning trees of I(n,k,l) in terms of Chebyshev polynomials. We investigate some arithmetical properties of this number and its asymptotic behaviour.

In 2009, Brown gave a set of conditions which when satisfied imply that a Feynman integral evaluates to a multiple zeta value. One of these conditions is called reducibility, which loosely says there is an order of integration for the Feynman integral for which Brown's techniques will succeed. Reducibility can be abstracted away from the Feynman integral to just being a condition on two polynomials, the first and second Symanzik polynomials. These polynomials can be defined from graphs, and thus reducibility is a property of graphs. We prove that for a fixed number of external momenta and no masses, reducibility is graph minor closed, correcting the previously claimed proofs of this fact. A computational study of reducibility was undertaken by Bogner and L\"{u}ders who found that for graphs with $4$-on-shell momenta and no masses, $K_{4}$ with momenta on each vertex is a forbidden minor. We add to this and find that when we restrict to graphs with four on-shell external momenta the following graphs are forbidden minors: $K_{4}$ with momenta on each vertex, $W_{4}$ with external momenta on the rim vertices, $K_{2,4}$ with external momenta on the large side of the bipartition, and one other graph. We do not expect that these minors characterize reducibility, so instead we give structural characterizations of the graphs not containing subsets of these minors. We characterize graphs not containing a rooted $K_{4}$ or rooted $W_{4}$ minor, graphs not containing rooted $K_{4}$ or rooted $W_{4}$ or rooted $K_{2,4}$ minors, and also a characterization of graphs not containing all of the known forbidden minors. Some comments are made on graphs not containing $K_{3,4}$, $K_{6}$ or a graph related to Wagner's graph as a minor.

This paper consists of three parts. First, we generalize the Jaeger Formula to express the Kauffman-Vogel graph polynomial as a state sum of the Murakami-Ohtsuki-Yamada graph polynomial. Then, we demonstrate that reversing the orientation and the color of a MOY graph along a simple circuit does not change the sl(N) Murakami-Ohtsuki-Yamada polynomial or the sl(N) homology of this MOY graph. In fact, reversing the orientation and the color of a component of a colored link only changes the sl(N) homology by an overall grading shift. Finally, as an application of the first two parts, we prove that the so(6) Kauffman polynomial is equal to the 2-colored sl(4) Reshetikhin-Turaev link polynomial, which implies that the 2-colored sl(4) link homology categorifies the so(6) Kauffman polynomial.

We investigate the relationship between the universal topological polynomials for graphs in mathematics and the parametric representation of Feynman amplitudes in quantum field theory. In this first paper we consider translation invariant theories with the usual heat-kernel-based propagator. We show how the Symanzik polynomials of quantum field theory are particular multivariate versions of the Tutte polynomial, and how the new polynomials of noncommutative quantum field theory are particular versions of the Bollob\'as-Riordan polynomials.

This paper surveys a comprehensive, although not exhaustive, sampling of graph polynomials with the goal of providing a brief overview of a variety of techniques defining a graph polynomial and then for decoding the combinatorial information it contains. The polynomials we discuss here are not generally specializations of the Tutte polynomial, but they are each in some way related to the Tutte polynomial, and often to one another. We emphasize these interrelations and explore how an understanding of one polynomial can guide research into others. We also discuss multivariable generalizations of some of these polynomials and the theory facilitated by this. We conclude with two examples, one from biology and one from physics, that illustrate the applicability of graph polynomials in other fields. This is the second chapter of a two chapter series, and concludes Graph Polynomials and Their Applications I: The Tutte Polynomial, arXiv:0803.3079

The Harborth graph is the smallest known example of a 4-regular planar unit-distance graph. In this paper we give an analytical description of the coordinates of its vertices for a particular embedding in the Euclidean plane. More precisely, we show, how to calculate the minimal polynomials of the coordinates of its vertices (with the help of a computer algebra system), and list those. Furthermore some algebraic properties of these polynomials, and consequences to the structure of the Harborth graph are determined.

Recently, M.\ Ab\'ert and T.\ Hubai studied the following problem. The chromatic measure of a finite simple graph is defined to be the uniform distribution on its chromatic roots. Ab\'ert and Hubai proved that for a Benjamini-Schramm convergent sequence of finite graphs, the chromatic measures converge in holomorphic moments. They also showed that the normalized log of the chromatic polynomial converges to a harmonic real function outside a bounded disc. In this paper we generalize their work to a wide class of graph polynomials, namely, multiplicative graph polynomials of bounded exponential type. A special case of our results is that for any fixed complex number $v_0$ the measures arising from the Tutte polynomial $Z_{G_n}(z,v_0)$ converge in holomorphic moments if the sequence $(G_n)$ of finite graphs is Benjamini--Schramm convergent. This answers a question of Ab\'ert and Hubai in the affirmative. Even in the original case of the chromatic polynomial, our proof is considerably simpler.

The quantum Bruhat graph, which is an extension of the graph formed by covering relations in the Bruhat order, is naturally related to the quantum cohomology ring of G/B. We enhance a result of Fulton and Woodward by showing that the minimal monomial in the quantum parameters that occurs in the quantum product of two Schubert classes has a simple interpretation in terms of directed paths in this graph. We define path Schubert polynomials, which are quantum cohomology analogues of skew Schubert polynomials recently introduced by Lenart and Sottile. They are given by sums over paths in the quantum Bruhat graph of type A. The 3-point Gromov-Witten invariants for the flag manifold are expressed in terms of these polynomials. This construction gives a combinatorial description for the set of all monomials in the quantum parameters that occur in the quantum product of two Schubert classes.

We introduce Macdonald characters and use algebraic properties of Macdonald polynomials to study them. As a result, we produce several formulas for Macdonald characters, which are generalizations of those obtained by Gorin-Panova in arXiv:1301.0634 [math.PR], and are expected to provide tools for the study of statistical mechanical models, representation theory and random matrices. As first application of our formulas, we characterize the boundary of the $(q, t)-$deformation of the Gelfand-Tsetlin graph.

Let $G$ be a simple $r$-regular graph with $n$ vertices and $m$ vertices. We give the signless Laplacian characteristic polynomials of $xyz$-transformations $G^{xyz}$ of $G$ in terms of $n$, $m$, $r$ and the signless Laplacian spectrum of $G$.

In this paper, we find computational formulae for generalized characteristic polynomials of graph bundles. We show that the number of spanning trees in a graph is the partial derivative (at (0,1)) of the generalized characteristic polynomial of the graph. Since the reciprocal of the Bartholdi zeta function of a graph can be derived from the generalized characteristic polynomial of a graph, consequently, the Bartholdi zeta function of a graph bundle can be computed by using our computational formulae.

In this paper, we find computational formulae for generalized characteristic polynomials of graph bundles. We show that the number of spanning trees in a graph is the partial derivative (at (0,1)) of the generalized characteristic polynomial of the graph. Since the reciprocal of the Bartholdi zeta function of a graph can be derived from the generalized characteristic polynomial of a graph, consequently, the Bartholdi zeta function of a graph bundle can be computed by using our computational formulae.

The appearance of multiple zeta values in anomalous dimensions and $\beta$-functions of renormalizable quantum field theories has given evidence towards a motivic interpretation of these renormalization group functions. In this paper we start to hunt the motive, restricting our attention to a subclass of graphs in four dimensional scalar field theory which give scheme independent contributions to the above functions.

Asymptotic expansions of Gaussian integrals may often be interpreted as generating functions for certain combinatorial objects (graphs with additional data). In this article we discuss a general approach to all such cases using colored graphs. We prove that the generating power series for such graphs satisfy the same system of partial differential equations as the Gaussian integral and the formal power series solution of this system is unique. The solution is obtained as the genus expansion of the generating power series. The initial term of this expansion is the corresponding generating function for trees. The consequence equations for this term turns to be equivalent to the inversion problem for the gradient mapping defined by the initial condition. The equations for the higher terms of the genus expansion are linear. The solutions of these equations can be expressed explicitly by substitution of the initial conditions and the initial term (the tree expansion) into some universal polynomials (for g>1) which are generating functions for stable closed graphs. (For g=1 instead of polynomials appears logarithm.) The stable graph polynomials satisfy certain recurrence. In [1] some of these results were obtained for the one dimensional case by more or less direct solution of differential equations. Here we present purely combinatorial proofs.

Many graph polynomials, such as the Tutte polynomial, the interlace polynomial and the matching polynomial, have both a recursive definition and a defining subset expansion formula. In this paper we present a general, logic-based framework which gives a precise meaning to recursive definitions of graph polynomials. We then prove that in this framework every recursive definition of a graph polynomial can be converted into a subset expansion formula.

In this paper, we present a method for characterizing the evolution of time-varying complex networks by adopting a thermodynamic representation of network structure computed from a polynomial (or algebraic) characterization of graph structure. Commencing from a representation of graph structure based on a characteristic polynomial computed from the normalized Laplacian matrix, we show how the polynomial is linked to the Boltzmann partition function of a network. This allows us to compute a number of thermodynamic quantities for the network, including the average energy and entropy. Assuming that the system does not change volume, we can also compute the temperature, defined as the rate of change of entropy with energy. All three thermodynamic variables can be approximated using low-order Taylor series that can be computed using the traces of powers of the Laplacian matrix, avoiding explicit computation of the normalized Laplacian spectrum. These polynomial approximations allow a smoothed representation of the evolution of networks to be constructed in the thermodynamic space spanned by entropy, energy, and temperature. We show how these thermodynamic variables can be computed in terms of simple network characteristics, e.g., the total number of nodes and node degree statistics for nodes connected by edges. We apply the resulting thermodynamic characterization to real-world time-varying networks representing complex systems in the financial and biological domains. The study demonstrates that the method provides an efficient tool for detecting abrupt changes and characterizing different stages in network evolution.

We define a new topological polynomial extending the Bollobas-Riordan one, which obeys a four-term reduction relation of the deletion/contraction type and has a natural behavior under partial duality. This allows to write down a completely explicit combinatorial evaluation of the polynomials, occurring in the parametric representation of the non-commutative Grosse-Wulkenhaar quantum field theory. An explicit solution of the parametric representation for commutative field theories based on the Mehler kernel is also provided.

We give a new characterization of the peak subalgebra of the algebra of quasisymmetric functions and use this to construct a new basis for this subalgebra. As an application of these results we obtain a combinatorial formula for the Kazhdan-Lusztig polynomials which holds in complete generality and is simpler and more explicit than any existing one. We then show that, in a certain sense, this formula cannot be simplified.

The graph isomorphism problem is considered. We assign modified characteristic polynomials for graphs and reduce the graph isomorphism problem to the following one. It is required to find out, is there such an enumeration of the graphs vertices that the polynomials of the graphs are equal. We present algorithms for the graph isomorphism problem that use the reduction. We prove the propositions that justify the possibility of a numerical realization of the algorithms for the general case of the graph isomorphism problem. The algorithms perform equality checking of graphs modified $n$-variables characteristic polynomials. We show that probability of obtaining a wrong solution of the graph isomorphism problem using recursive modification of the algorithm is negligible if the algorithm parameter is sufficiently large. In the course of its implementation, the algorithm checks the equality of the graphs modified characteristic polynomials in predefined points. For $n$-vertices graph, the polynomial has $2^n$ coefficients so its value in some point cannot be evaluated directly for large enough $n$. We show that we may check the equality of the polynomials in predefined points without direct evaluation of the polynomials values in these points. We prove that, for the graphs on $n$ vertices, it is required $O(n^4)$ elementary machine operations and it is requred machine numbers with mantissa's length $O(n^2)$ to check equality of the graphs polynomials values in predefined points. In general, it needs an exponential time to solve the $GI$ instance using the presented approach, but in practice, it is efficient even for compuationally hard instances of the graph isomorphism problem.

We propose a duality analysis for obtaining the critical manifold of two-dimensional spin glasses. Our method is based on the computation of quenched free energies with periodic and twisted periodic boundary conditions on a finite basis. The precision can be systematically improved by increasing the size of the basis, leading to very fast convergence towards the thermodynamic limit. We apply the method to obtain the phase diagrams of the random-bond Ising model and $q$-state Potts gauge glasses. In the Ising case, the Nishimori point is found at $p_N = 0.10929 \pm 0.00002$, in agreement with and improving on the precision of existing numerical estimations. Similar precision is found throughout the high-temperature part of the phase diagram. Finite-size effects are larger in the low-temperature region, but our results are in qualitative agreement with the known features of the phase diagram. In particular we show analytically that the critical point in the ground state is located at finite $p_0$.

The Bethe approximation, or loopy belief propagation algorithm is a successful method for approximating partition functions of probabilistic models associated with a graph. Chertkov and Chernyak derived an interesting formula called Loop Series Expansion, which is an expansion of the partition function. The main term of the series is the Bethe approximation while other terms are labeled by subgraphs called generalized loops. In our recent paper, we derive the loop series expansion in form of a polynomial with coefficients positive integers, and extend the result to the expansion of marginals. In this paper, we give more clear derivation for the results and discuss the properties of the polynomial which is introduced in the paper.

The critical curves of the q-state Potts model can be determined exactly for regular two-dimensional lattices G that are of the three-terminal type. Jacobsen and Scullard have defined a graph polynomial P_B(q,v) that gives access to the critical manifold for general lattices. It depends on a finite repeating part of the lattice, called the basis B, and its real roots in the temperature variable v = e^K - 1 provide increasingly accurate approximations to the critical manifolds upon increasing the size of B. These authors computed P_B(q,v) for large bases (up to 243 edges), obtaining determinations of the ferromagnetic critical point v_c > 0 for the (4,8^2), kagome, and (3,12^2) lattices to a precision (of the order 10^{-8}) slightly superior to that of the best available Monte Carlo simulations. In this paper we describe a more efficient transfer matrix approach to the computation of P_B(q,v) that relies on a formulation within the periodic Temperley-Lieb algebra. This makes possible computations for substantially larger bases (up to 882 edges), and the precision on v_c is hence taken to the range 10^{-13}. We further show that a large variety of regular lattices can be cast in a form suitable for this approach. This includes all Archimedean lattices, their duals and their medials. For all these lattices we tabulate high-precision estimates of the bond percolation thresholds p_c and Potts critical points v_c. We also trace and discuss the full Potts critical manifold in the (q,v) plane, paying special attention to the antiferromagnetic region v < 0. Finally, we adapt the technique to site percolation as well, and compute the polynomials P_B(p) for certain Archimedean and dual lattices (those having only cubic and quartic vertices), using very large bases (up to 243 vertices). This produces the site percolation thresholds p_c to a precision of the order 10^{-9}.

We describe in this talk three methods of constructing different links with the same Jones type invariant. All three can be thought as generalizations of mutation. The first combines the satellite construction with mutation. The second uses the notion of rotant, taken from the graph theory, the third, invented by Jones, transplants into knot theory the idea of the Yang-Baxter equation with the spectral parameter (idea employed by Baxter in the theory of solvable models in statistical mechanics). We extend the Jones result and relate it to Traczyk's work on rotors of links. We also show further applications of the Jones idea, e.g. to 3-string links in the solid torus. We stress the fact that ideas coming from various areas of mathematics (and theoretical physics) has been fruitfully used in knot theory, and vice versa. (This is the detailed version of the talk given at the Banach Center Colloquium, Warsaw, Poland, March 24, 1994: ``W poszukiwaniu nietrywialnego wezla z trywialnym wielomianem Jonesa: grafy i mechanika statystyczna")

Using Jones correspondence between elements in the Thompson group F and certain graphs/links we establish, for certain specializations of the variables, the positive definiteness of some familiar link invariants and graph polynomials, namely the Kauffman bracket, the number of n-colourings and the Tutte polynomial, when viewed as functions on F.

We study the dual graph polynomials and the case when a Feynman graph has no triangles but has a 4-face. This leads to the proof of the duality-admissibility of all graphs up to 18 loops. As a consequence, the $c_2$ invariant is the same for all 4 Feynman period representations (position, momentum, parametric and dual parametric) for any physically relevant graph.

In recent years enormous progress has been made in perturbative quantum field theory by applying methods of algebraic geometry to parametric Feynman integrals for scalar theories. The transition to gauge theories is complicated not only by the fact that their parametric integrand is much larger and more involved. It is, moreover, only implicitly given as the result of certain differential operators applied to the scalar integrand $\exp(-\Phi_{\Gamma}/\Psi_{\Gamma})$, where $\Psi_{\Gamma}$ and $\Phi_{\Gamma}$ are the Kirchhoff and Symanzik polynomials of the Feynman graph $\Gamma$. In the case of quantum electrodynamics we find that the full parametric integrand inherits a rich combinatorial structure from $\Psi_{\Gamma}$ and $\Phi_{\Gamma}$. In the end, it can be expressed explicitly as a sum over products of new types of graph polynomials which have a combinatoric interpretation via simple cycle subgraphs of $\Gamma$.

We introduce two graph polynomials and discuss their properties. One is a polynomial of two variables whose investigation is motivated by the performance analysis of the Bethe approximation of the Ising partition function. The other is a polynomial of one variable that is obtained by the specialization of the first one. It is shown that these polynomials satisfy deletion-contraction relations and are new examples of the V-function, which was introduced by Tutte (1947, Proc. Cambridge Philos. Soc. 43, 26-40). For these polynomials, we discuss the interpretations of special values and then obtain the bound on the number of sub-coregraphs, i.e., spanning subgraphs with no vertices of degree one. It is proved that the polynomial of one variable is equal to the monomer-dimer partition function with weights parameterized by that variable. The properties of the coefficients and the possible region of zeros are also discussed for this polynomial.

Graph polynomials are graph parameters invariant under graph isomorphisms which take values in a polynomial ring with a fixed finite number of indeterminates. We study graph polynomials from a model theoretic point of view. In this paper we distinguish between the graph theoretic (semantic) and the algebraic (syntactic) meaning of graph polynomials. We discuss how to represent and compare graph polynomials by their distinctive power. We introduce the class of graph polynomials definable using Second Order Logic which comprises virtually all examples of graph polynomials with a fixed finite set of indeterminates. Finally we show that the location of zeros and stability of graph polynomials is not a semantic property. The paper emphasizes a model theoretic view and gives a unified exposition of classical results in algebraic combinatorics together with new and some of our previously obtained results scattered in the graph theoretic literature.

We prove optimality of the Arf invariant formula for the generating function of even subgraphs, or, equivalently, the Ising partition function, of a graph.

We find families of prime knot diagrams with arbitrary extreme coefficients in their Jones polynomials. Some graph theory is presented in connection with this problem, generalizing ideas by Yongju Bae and Morton and giving a positive answer to a question in their paper.

This paper deals with crtain posets and lattices associated with a graph. In my earlier paper I showed that several invariants of a graph can be computed from the isomorphism class of its poset of non-empty induced subgraphs. In this paper I will prove that the (abstract and folded) connected partition lattice of a graph can be constructed from its abstract poset of induced subgraphs. I will also prove that, except when the graph is a star or a disjoint union of edges, the abstract induced subgraph poset of the graph can be constructed from its abstract folded connected partition lattice. The chromatic symmetric function and the symmetric Tutte polynomial are proved to be reconstructible. The second construction implies that a tree can be reconstructed from the isomorphism class of its folded connected partition lattice. I then show that the symmetric Tutte polynomial of a tree can be computed from the chromatic symmetric function of the tree, thus showing that a question of Noble and Welsh is equivalent to Stanley's question about the chromatic symmetric function of trees. The paper also develops edge reconstruction theory on the edge subgraph poset, and its relation with Lov\'asz's homomorphism cancellation laws. A characterisation of a family of graphs that cannot be constructed from their abstract edge subgraph posets is also presented.

Graph polynomials are polynomials assigned to graphs. Interestingly, they also arise in many areas outside graph theory as well. Many properties of graph polynomials have been widely studied. In this paper, we survey some results on the derivative and real roots of graph polynomials, which have applications in chemistry, control theory and computer science. Related to the derivatives of graph polynomials, polynomial reconstruction of the matching polynomial is also introduced.

Averbouch, Godlin and Makowsky define the edge elimination polynomial of a graph by a recurrence relation with respect to the deletion, contraction and extraction of an edge. It generalizes some well-known graph polynomials such as the chromatic polynomial and the matching polynomial. By introducing two equivalent graph polynomials, one enumerating subgraphs and the other enumerating colorings, we show that the edge elimination polynomial of a simple graph is reconstructible from its polynomial deck and that it encodes the degree sequence of an arbitrary graph.

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.

In this survey of graph polynomials, we emphasize the Tutte polynomial and a selection of closely related graph polynomials. We explore some of the Tutte polynomial's many properties and applications and we use the Tutte polynomial to showcase a variety of principles and techniques for graph polynomials in general. These include several ways in which a graph polynomial may be defined and methods for extracting combinatorial information and algebraic properties from a graph polynomial. We also use the Tutte polynomial to demonstrate how graph polynomials may be both specialized and generalized, and how they can encode information relevant to physical applications. We conclude with a brief discussion of computational complexity considerations.

In this survey of graph polynomials, we emphasize the Tutte polynomial and a selection of closely related graph polynomials. We explore some of the Tutte polynomial's many properties and applications and we use the Tutte polynomial to showcase a variety of principles and techniques for graph polynomials in general. These include several ways in which a graph polynomial may be defined and methods for extracting combinatorial information and algebraic properties from a graph polynomial. We also use the Tutte polynomial to demonstrate how graph polynomials may be both specialized and generalized, and how they can encode information relevant to physical applications. We conclude with a brief discussion of computational complexity considerations.

In the present paper we find a simple algorithm for counting Jacobian group of the generalized Petersen graph GP(n,k). Also, we obtain a closed formula for the number of spanning trees of this graph in terms of Chebyshev polynomials.

In Chapter 2 we study the path-cycle symmetric function of a digraph, a symmetric function generalization of Chung and Graham's cover polynomial. Most of this material appears in either Advances in Math. 118 (1996), 71-98 or J. Algebraic Combin. 10 (1999), 227-240. Chapter 3 contains miscellaneous results about Stanley's symmetric function generalization X_G of the chromatic polynomial, e.g., we establish a connection with some of Tutte's work on the chromatic polynomial and use this to prove that X_G is reconstructible. Most of Chapter 3 does not appear elsewhere.

In this paper, we will show dichotomy theorems for the computation of polynomials corresponding to evaluation of graph homomorphisms in Valiant's model. We are given a fixed graph $H$ and want to find all graphs, from some graph class, homomorphic to this $H$. These graphs will be encoded by a family of polynomials. We give dichotomies for the polynomials for cycles, cliques, trees, outerplanar graphs, planar graphs and graphs of bounded genus.

We report exact calculations of the ground state degeneracy per site (exponent of the ground state entropy) $W(\{G\},q)$ of the $q$-state Potts antiferromagnet on infinitely long strips with specified end graphs for free boundary conditions in the longitudinal direction and free and periodic boundary conditions in the transverse direction. This is equivalent to calculating the chromatic polynomials and their asymptotic limits for these graphs. Making the generalization from $q \in {\mathbb Z}_+$ to $q \in {\mathbb C}$, we determine the full locus ${\cal B}$ on which $W$ is nonanalytic in the complex $q$ plane. We report the first example for this class of strip graphs in which ${\cal B}$ encloses regions even for planar end graphs. The bulk of the specific strip graph that exhibits this property is a part of the $(3^3 \cdot 4^2)$ Archimedean lattice.

Correlation functions in quantum field theory are calculated using Feynman amplitudes, which are finite dimensional integrals associated to graphs. The integrand is the exponential of the ratio of the first and second Symanzik polynomials associated to the Feynman graph, which are described in terms of the spanning trees and spanning 2-forests of the graph, respectively. In a previous paper with Bloch, Burgos and Fres\'an, we related this ratio to the asymptotic of the Archimedean height pairing between degree zero divisors on degenerating families of Riemann surfaces. Motivated by this, we consider in this paper the variation of the ratio of the two Symanzik polynomials under bounded perturbations of the geometry of the graph. This is a natural problem in connection with the theory of nilpotent and SL2 orbits in Hodge theory. Our main result is the boundedness of variation of the ratio. For this we define the exchange graph of a given graph which encodes the exchange properties between spanning trees and spanning 2-forests in the graph. We provide a description of the connected components of this graph, and use this to prove our result on boundedness of the variations.

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), and a description of a family of polynomials containing the graph polynomial which yield Tutte-Grothendieck invariants in a similar way.

In this talk I discuss properties of the two Symanzik polynomials which characterise the integrand of an arbitrary multi-loop integral in its Feynman parametric form. Based on the construction from spanning forests and Laplacian matrices, Dodgson's relation is applied to derive factorisation identities involving both polynomials. An application of Whitney's 2-isomorphism theorem on matroids is discussed.

The integrand of any multi-loop integral is characterised after Feynman parametrisation by two polynomials. In this review we summarise the properties of these polynomials. Topics covered in this article include among others: Spanning trees and spanning forests, the all-minors matrix-tree theorem, recursion relations due to contraction and deletion of edges, Dodgson's identity and matroids.

Graph polynomials are deemed useful if they give rise to algebraic characterizations of various graph properties, and their evaluations encode many other graph invariants. Algebraic: The complete graphs $K_n$ and the complete bipartite graphs $K_{n,n}$ can be characterized as those graphs whose matching polynomials satisfy a certain recurrence relations and are related to the Hermite and Laguerre polynomials. An encoded graph invariant: The absolute value of the chromatic polynomial $\chi(G,X)$ of a graph $G$ evaluated at $-1$ counts the number of acyclic orientations of $G$. In this paper we prove a general theorem on graph families which are characterized by families of polynomials satisfying linear recurrence relations. This gives infinitely many instances similar to the characterization of $K_{n,n}$. We also show where to use, instead of the Hermite and Laguerre polynomials, linear recurrence relations where the coefficients do not depend on $n$. Finally, we discuss the distinctive power of graph polynomials in specific form.