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Catalogue of graph polynomials
... Comme perspectives, nous comptonsétudier d'autres polynômes de graphes (voir par exemple[74]), notamment le polynôme de Tutte[99]. Nous envisageons aussi d'étudier le polynôme chromatiqueà l'aide d'autres structures discrètes, tels que les treillis, les complexes simpliciaux et les matroïdes.Graphe de Γ d'ordre n représenté par d. chrom gamma(d) Vecteur d du mot associé a un graphe de Γ d'ordre n. chromatique du graphe de Γ d'ordre n représenté par b. ...
Some subclasses of chrodal graphs that are auto-complementary, i.e their complementary belong to the same subclass. Our main contribution concerns the chromatic polynomials of threshold graphs, that are auto-complementary, for which we have find several algebraic and combinatorial properties.
Besides, we have defined a generalization of threshold graphs that exhibits interesting combinatorial properties.
Keyword. proper coloration; chordal graphs; threshold graphs; complementary graphs; generalized Bell numbers; chromatic polynomial.
... It is easy to find by Lemma 3.2.2, taking the center of the star as our distinguished vertex v, that s n (x) = x+(1+x) n−1 ; see, e.g., [38]. Thus, by Proposition 3.3.1, ...
Given two graphs G and H, a homomorphism from G to H is a map from the vertices of G to the vertices of H that preserves adjacency. Many graph notions can be described using graph homomorphisms, including independent sets, matchings, and graph colorings. In this dissertation, we consider questions in each of these areas.
Wang and Zhu demonstrated the log-concavity of the independence polynomial for a number of infinite recursively-defined graph classes. For the first question that we consider, we extend their method to demonstrate the log-concavity of the independence polynomial for a wider range of such classes, showing, for example, that the (n; 2)-centipede and (n; 2; k)-star-centipede have log-concave independence polynomials for even n and all k, and we describe a technique that can be applied in the pursuit of results for other similarly-structured graphs. The graphs in question give rise to k-periodic recurrence relations. We present a system for reducing these periodic recurrence relations to ordinary second-order recurrence relations that is more applicable than existing work in the field.
There has been much discussion on the following question: for each fixed H, n and d, what is the maximum, over all n-vertex, d-regular G, of hom(G;H), the number of homomorphisms from G to H? While the question has been largely settled for bipartite G, it is still quite open for non-bipartite G. Offering further progress on this question, we next establish conditions on H, in terms of certain linear programming problems, under which we can obtain an upper bound on hom(G;H), valid for all regular G, that is very close to best possible. We then establish a number of infinite families of graphs that satisfy these conditions.
Finally, we present a q-weighted enumeration of matchings of complete bipartite graphs, an enumeration which was key to proving the validity of a new combinatorial interpretation of the q-analog of a generalization of the Stirling numbers of the second kind in work with Engbers and Galvin.
International audience
The problem of counting monomer-dimer coverings of a lattice is a longstanding problem in statistical mechanics.It has only been exactly solved for the special case of dimer coverings in two dimensions ([Ka61], [TF61]). In earlier work, Stanley [St85] proved a reciprocity principle governing the number N(m,n) of dimer coverings of an m by n rectangular grid (also known as perfect matchings), where m is fixed and n is allowed to vary. As reinterpreted by Propp [P01], Stanley's result concerns the unique way of extending N(m,n) to so that the resulting bi-infinite sequence, N(m,n) for , satisfies a linear recurrence relation with constant coefficients. In particular, Stanley shows that N(m,n) is always an integer satisfying the relation where unless and n is odd, in which case . Furthermore, Propp's method was applicable to higher-dimensional cases.This paper discusses similar investigations of the numbers M(m,n), of monomer-dimer coverings, or equivalently (not necessarily perfect) matchings of an m by n rectangular grid. We show that for each fixed m there is a unique way of extending M(m,n) to so that the resulting bi-infinite sequence, M(m,n) for , satisfies a linear recurrence relation with constant coefficients.We show that M(m,n), a priori a rational number, is always an integer, using a generalization of the combinatorial model offered by Propp. Lastly, we give a new statement of reciprocity in terms of multivariate generating functions from which Stanley's result follows.
This is the first book to comprehensively cover chromatic polynomials of graphs. It includes most of the known results and unsolved problems in the area of chromatic polynomials. Dividing the book into three main parts, the authors take readers from the rudiments of chromatic polynomials to more complex topics: the chromatic equivalence classes of graphs and the zeros and inequalities of chromatic polynomials. The early material is well suited to a graduate level course while the latter parts will be an invaluable resource for postgraduate students and researchers in combinatorics and graph theory. © 2005 by World Scientific Publishing Co. Pte. Ltd. All rights reserved.
Let G be a simple graph of order n and size m. An edge covering of a graph is a set of edges such that every vertex of the graph is incident to at least one edge of the set. In this paper we introduce a new graph polynomial. The edge cover polynomial of G is the polynomial E(G,x)=@?"i"="1^me(G,i)x^i, where e(G,i) is the number of edge coverings of G of size i. Let G and H be two graphs of order n such that @d(G)>=n2, where @d(G) is the minimum degree of G. If E(G,x)=E(H,x), then we show that the degree sequence of G and H are the same. We determine all graphs G for which E(G,x)=E(P"n,x), where P"n is the path of order n. We show that if @d(G)>=3, then E(G,x) has at least one non-real root. Finally, we characterize all graphs whose edge cover polynomials have exactly one or two distinct roots and moreover we prove that these roots are contained in the set {-3,-2,-1,0}.
In this report we define a new coloring of graphs, namely harmonious coloring of graphs, which arises as an extension of harmonious and graceful numbering of graphs. We show that the harmonious coloring problem for general graphs is NP-complete.
A stable (or independent) set in a graph is a set of pairwise non-adjacent vertices. The stability number fi(G) is the size of a maximum stable set in the graph G. There are three dierent kinds of structures that one can see observing behavior of stable sets of a graph: the enumerative structure, the intersection structure, and the exchange structure. The independence polynomial of G I(G;x) =
We consider several generalizations of rook polynomials. In particular we develop analogs of the theory of rook polynomials that are related to general Laguerre and Charlier polynomials in the same way that ordinary rook polynomials are related to simple Laguerre polynomials.
A topological index Z is proposed for a connected graph G representing the carbon skeleton of a saturated hydrocarbon. The integer Z is the sum of a set of the numbers p(G,k), which is the number of ways in which such k bonds are so chosen from G that no two of them are connected. For chain molecules Z is closely related to the characteristic polynomial derived from the topological matrix. It is found that Z is correlated well with the topological nature of the carbon skeleton, i.e., the mode of branching and ring closure. Some interesting relations are found, such as a graphical representation of the Fibonacci numbers and a composition principle for counting Z. Correlation of Z with boiling points of saturated hydrocarbons is pointed out.
This graduate level text is distinguished both by the range of topics and the novelty of the material it treats-more than half of the material in it has previously only appeared in research papers. The first half of this book introduces the characteristic and matchings polynomials of a graph. It is instructive to consider these polynomials together because they have a number of properties in common. The matchings polynomial has links with a number of problems in combinatorial enumeration, particularly some of the current work on the combinatorics of orthogonal polynomials. This connection is discussed at some length, and is also in part the stimulus for the inclusion of chapters on orthogonal polynomials and formal power series. Many of the properties of orthogonal polynomials are derived from properties of characteristic polynomials. The second half of the book introduces the theory of polynomial spaces, which provide easy access to a number of important results in design theory, coding theory and the theory of association schemes. This book should be of interest to second year graduate text/reference in mathematics.
We introduce a new tool, the factorial polynomials, to study rook equivalence of Ferrers boards. We provide a set of invariants for rook equivalence as well as a very simple algorithm for deciding rook equivalence of Ferrers boards. We then count the number of Ferrers boards rook equivalent to a given Ferrers board.
We have considered a two-dimensional square net consisting of four kinds of atoms supposing that only nearest neighbors interact and that there are only two distinct potential energies of interaction, one between like and one between unlike atoms. In extension of a method due to Onsager it is found that for the case where like atoms attract one another a simple "reciprocity" relation exists between the partition functions at pairs of temperatures "reciprocally" related to one another. As one temperature T tends to zero, the other T* tends to infinity. If one further assumes that only one "Curie" transition point exists, the relation between T and T* enables one to locate the Curie temperature. Predictions can be made concerning the nature of the transition point with results similar to those of Kramers and Wannier. The reciprocity relation for the case of attraction between like atoms is found to be not valid for the case where unlike atoms attract one another.
Chemistry and graph theory meet in several areas which are briefly reviewed. A few solved and unsolved problems are discussed: generalized centers in cyclic graphs; irreducible sequences in polymers; cages; spectral graph theoretical problems; k-factorable graphs with k>1, and perfect matchings with k=1.
A rook polynomial is a polynomial whose x
k
coefficient is the number of ways k rooks can be placed on the squares of an arbitrarily shaped chessboard so that no rooks share the same rows or columns. The k rooks are called non-taking. Rook polynomials pattern combinatorial situations, especially those involving restricted permutations. The conventional square board used in the game, chess, is but one configuration.
The harmonious chromatic number of a graph is the least number of colours in a vertex colouring such that each pair of colours appears on at most one edge. The achromatic number of a graph is the greatest number of colours in a vertex colouring such that each pair of colours appears on at least one edge. This paper is a survey of what is known about these two parameters, in particular we look at upper and lower bounds, special classes of graphs and complexity issues.
We show that given an Eulerian directed graph G there is a polynomial m(G;ζ) in one variable with non-negative integer coefficients satisfying certain inductive relations, such that m(G; 1) is the number of Eulerian circuits of G. This solves a conjecture due to Martin. Similar results hold for undirected Eulerian graphs and 4-regular graphs drawn on surfaces. Using properties of these polynomials we derive bounds for the number of Eulerian orientations of an undirected graph with even degrees.
This chapter reviews graph coloring problems at different levels of the complexity hierarchy and discusses their use to illustrate various complexity levels within the languages of graphs. The complexity terminology is fairly standard and follows closely that used by Garey and Johnson. The first problem of testing whether a graph is 2-colorable is introduced principally to clarify the notion space in nondeterministic machines. The second set of problems concern the number of colorings of a graph. The chapter discusses the creation of distinct recognition (= decision) problems associated with the well-known problem of finding the chromatic polynomial of a graph. Testing whether or not a graph is 2-colourable is one of the easiest algorithmic problems in the graph theory. The chromatic polynomial is also reviewed in the chapter.
For a graph G, we denote by P(G, λ) the chromatic polynomial of G and by h(G, x) the adjoint polynomial of G. A graph G is said to be chromatically unique if for any graph H, P(H, λ) = P(G, λ) implies H ≅ G. In this paper, we investigate some algebraic properties of the adjoint polynomials of some graphs. Using these properties, we obtain necessary and sufficient conditions for Kn - E(∪a,bT1,a,b) and (∪iCni) ∪ (∪iDmj) ∪ (∪a,bT1,a,b) to be chromatically unique if Gi ∈ {Cn,Dn, T1,a,b|n ≥ 5, 3 ≤ a ≤ 10, a ≤ b} and h(Pm) h(Gi) for all m ≥ 2. Moreover, many new chromatically unique graphs are given.
For a finite graph G with d vertices we define a homogeneous symmetric function XG of degree d in the variables x1, x2, ... . If we set x1 = ... = xn= 1 and all other xi = 0, then we obtain χG(n), the chromatic polynomial of G evaluated at n. We consider the expansion of XG in terms of various symmetric function bases. The coefficients in these expansions are related to partitions of the vertices into stable subsets, the Möbius function of the lattice of contractions of G, and the structure of the acyclic orientations of G. The coefficients which arise when XG is expanded in terms of elementary symmetric functions are particularly interesting, and for certain graphs are related to the theory of Hecke algebras and Kazhdan-Lusztig polynomials.
Two bipartite graphs and in which there are no isolated points and in which the cardinalities of the ‘upper’ sets are equal, that is, |S1|=|S2|=n (say), are said to be matching-equivalent if and only if the number of r-matchings (i.e., the number of ways in which r disjoint edges can be chosen) is the same for each of the graphs G1 and G2 for each r,1⩽r⩽n. We show that the number of bipartite graphs that are matching-equivalent to , the complete bipartite graph of order (n,n) is 2n−1 subject to an inclusion condition on the sets of neighbors vertices of the ‘upper set’. The proof involves adding an arbitrary number of vertices to the ‘lower’ set which are neighbors to all the vertices in the upper set and then analyzing the ‘modified’ rook polynomial that is specially defined for the purpose of the proof.
In this paper we define the vertex-cover polynomial Ψ(G,τ) for a graph G. The coefficient of τr in this polynomial is the number of vertex covers V′ of G with |V′|=r. We develop a method to calculate Ψ(G,τ). Motivated by a problem in biological systematics, we also consider the mappings f from {1,2,…,m} into the vertex set V(G) of a graph G, subject to f−1(x)∪f−1(y)≠∅ for every edge xy in G. Let F(G,m) be the number of such mappings f. We show that F(G,m) can be determined from Ψ(G,τ).
Polygraphs are introduced in order to describe and generalize the chemical notion of polymers. A general method for determining the matching polynomial of a polygraph is presented.
The number of ways of placingknon-attacking rooks on a Ferrers board is expressed as a hypergeometric series of a type originally studied by Karlsson (J. Math. Phys.12(1971), 270–271) and Minton (J. Math. Phys.11(1970), 1375–1376). Known transformation identities for series of this type translate into new theorems about rook polynomials.
We explain combinatorially the occurrence of certain classical sequences of orthogonal polynomials as sequences of rook polynomials, and we give some new examples related to general stairstep boards.
We explore the relation between rook theory of Ferrers boards and polynomial sequences of binomial type. Recursion and explicit formulas for the rook numbers of Riordan's trapezoidal boards are derived. It is shown that some classic sequences of rook and factorial polynomials are of binomial type. A class of boards corresponding to Abel's theorem and its generalizations are constructed. The results are developed from both algebraic and combinatorial viewpoints. Introduction. In (1) we began the algebraic study of the rook and the factorial polynomials and the classification and enumeration of rook equivalent Ferrers boards. The purpose of this paper is to develop the intimate connection of our theory with the theory of polynomial sequences of binomial types (2), (4) and the more general theory of Sheffer sequences. The paper is divided into two parts, the algebraic theory and the combinator- ial theory. Section 1 reviews some. basic notions of polynomial sequences of binomial type. In ? 1.2 we introduce the notion of a sequence of boards of binomial type and prove a classification theorem. In ?? 1.3 and 1.4 we solve the classic problems raised by Riordan (3, Chap. 8) for trapezoidal boards, namely, we derive recursions and explicit formulas for the rook numbers and exponential generating functions for sequences of rook polynomials. We also show that certain classic sequences of rook and factorial polynomials are of binomial type, a quite
The spectra of matching polynomials which are useful in the computations of resonance energy and grand canonical partition functions and other properties are obtained for certain classes of graphs and lattices. All the eigenvalues are obtainable for graphs which possess Hermitian adjacency matrices whose secular determinants are the matching polynomials. Several illustrative examples are provided.
We present an innite family of 3-connected non-bipartite graphs with chromatic roots in the interval (1; 2) thus resolving a conjecture of Jackson's in the negative. In addition, we briey consider other graph classes that are conjectured to have no chromatic roots in (1; 2).
Motivated by the work of Chmutov, Duzhin and Lando on Vassiliev invariants, we define a polynomial on weighted graphs which contains as specialisations the weighted chromatic invariants but also contains many other classical invariants including the Tutte and matching polynomials. The paper also gives the symmetric function generalisation of the chromatic polynomial introduced by Stanley. We study its complexity and prove hardness results for very restricted classes of graphs.
Two polynomials θ ( G, n ) and ϕ ( G, n ) connected with the colourings of a graph G or of associated maps are discussed. A result believed to be new is proved for the lesser-known polynomial ϕ ( G, n ). Attention is called to some unsolved problems concerning ϕ ( G, n ) which are natural generalizations of the Four Colour Problem from planar graphs to general graphs. A polynomial χ ( G, x, y ) in two variables x and y , which can be regarded as generalizing both θ ( G, n ) and ϕ ( G, n ) is studied. For a connected graph χ ( G, x, y ) is defined in terms of the “spanning” trees of G (which include every vertex) and in terms of a fixed enumeration of the edges.