Milnor Numbers of Projective Hypersurfaces and the Chromatic Polynomial of Graphs
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Notes on the Topology of Complex Singularities
Notes on the Topology of Complex Singularities Liviu I. Nicolaescu University of Notre Dame, Indiana, USA Last modified: October 13, 2013. i Introduction The algebraic varieties have played a very important role in the development of geometry. The lines and the conics where the first to be investigated and moreover, the study of equations leads naturally to algebraic geometry. The past century has witnessed the introduction of new ideas and techniques, notably algebraic topology and complex geometry. These had a dramatic impact on the development of this subject. There are several reasons which make algebraic varieties so attractive. On one hand, it is their abundance and the wealth of techniques available to study them and, on the other hand, there are the often unexpected conclusions. These conclusions lead frequently to new research questions in other directions. The gauge theoretic revolution of the past two decades has only increased the role played by these objects. More recently, Simon Donaldson has drawn attention to Lefschetz’ old techniques of studying algebraic manifolds by extending them to the much more general context of symplectic manifolds. I have to admit that I was not familiar with Lefschetz’ ideas and this gave me the impetus to teach a course on this subject and write up semi- formal notes. The second raison d’ˆetre of these notes is my personal interest in the isolated singularities of complex surfaces. Loosely speaking, Lefschetz created a holomorphic version of Morse theory when the traditional one was not even born. He showed that a holomorphic map f from a complex manifold M to the complex projective line P1 which admits only nondegenerate critical points contains a large amount of nontrivial topological information about M. -
CHROMATIC POLYNOMIALS of PLANE TRIANGULATIONS 1. Basic
1990, 2005 D. R. Woodall, School of Mathematical Sciences, University of Nottingham CHROMATIC POLYNOMIALS OF PLANE TRIANGULATIONS 1. Basic results. Throughout this survey, G will denote a multigraph with n vertices, m edges, c components and b blocks, and m′ will denote the smallest number of edges whose deletion from G leaves a simple graph. The corresponding numbers for Gi will be ′ denoted by ni , mi , ci , bi and mi . Let P(G, t) denote the number of different (proper vertex-) t-colourings of G. Anticipating a later result, we call P(G, t) the chromatic polynomial of G. It was introduced by G. D. Birkhoff (1912), who proved many of the following basic results. Proposition 1. (Examples.) Here Tn denotes an arbitrary tree with n vertices, Fn denotes an arbitrary forest with n vertices and c components, and Rn denotes the graph of an arbitrary triangulated polygon with n vertices: that is, a plane n-gon divided into triangles by n − 3 noncrossing chords. = n (a) P(Kn , t) t , = − n −1 (b) P(Tn , t) t(t 1) , = − − n −2 (c) P(Rn , t) t(t 1)(t 2) , = − − − + (d) P(Kn , t) t(t 1)(t 2)...(t n 1), = c − n −c (e) P(Fn , t) t (t 1) , = − n + − n − (f) P(Cn , t) (t 1) ( 1) (t 1) − = (−1)nt(t − 1)[1 + (1 − t) + (1 − t)2 + ... + (1 − t)n 2]. Proposition 2. (a) If the components of G are G1,..., Gc , then = P(G, t) P(G1, t)...P(Gc , t). = ∪ ∩ = (b) If G G1 G2 where G1 G2 Kr , then P(G1, t) P(G2, t) ¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡£¡¢¡¢¡¢¡ P(G, t) = ¡ . -
Approximating the Chromatic Polynomial of a Graph
Approximating the Chromatic Polynomial Yvonne Kemper1 and Isabel Beichl1 Abstract Chromatic polynomials are important objects in graph theory and statistical physics, but as a result of computational difficulties, their study is limited to graphs that are small, highly structured, or very sparse. We have devised and implemented two algorithms that approximate the coefficients of the chromatic polynomial P (G, x), where P (G, k) is the number of proper k-colorings of a graph G for k ∈ N. Our algorithm is based on a method of Knuth that estimates the order of a search tree. We compare our results to the true chro- matic polynomial in several known cases, and compare our error with previous approximation algorithms. 1 Introduction The chromatic polynomial P (G, x) of a graph G has received much atten- tion as a result of the now-resolved four-color problem, but its relevance extends beyond combinatorics, and its domain beyond the natural num- bers. To start, the chromatic polynomial is related to the Tutte polynomial, and evaluating these polynomials at different points provides information about graph invariants and structure. In addition, P (G, x) is central in applications such as scheduling problems [26] and the q-state Potts model in statistical physics [10, 25]. The former occur in a variety of contexts, from algorithm design to factory procedures. For the latter, the relation- ship between P (G, x) and the Potts model connects statistical mechanics and graph theory, allowing researchers to study phenomena such as the behavior of ferromagnets. Unfortunately, computing P (G, x) for a general graph G is known to be #P -hard [16, 22] and deciding whether or not a graph is k-colorable is NP -hard [11]. -
Computing Tutte Polynomials
Computing Tutte Polynomials Gary Haggard1, David J. Pearce2, and Gordon Royle3 1 Bucknell University [email protected] 2 Computer Science Group, Victoria University of Wellington, [email protected] 3 School of Mathematics and Statistics, University of Western Australia [email protected] Abstract. The Tutte polynomial of a graph, also known as the partition function of the q-state Potts model, is a 2-variable polynomial graph in- variant of considerable importance in both combinatorics and statistical physics. It contains several other polynomial invariants, such as the chro- matic polynomial and flow polynomial as partial evaluations, and various numerical invariants such as the number of spanning trees as complete evaluations. However despite its ubiquity, there are no widely-available effective computational tools able to compute the Tutte polynomial of a general graph of reasonable size. In this paper we describe the implemen- tation of a program that exploits isomorphisms in the computation tree to extend the range of graphs for which it is feasible to compute their Tutte polynomials. We also consider edge-selection heuristics which give good performance in practice. We empirically demonstrate the utility of our program on random graphs. More evidence of its usefulness arises from our success in finding counterexamples to a conjecture of Welsh on the location of the real flow roots of a graph. 1 Introduction The Tutte polynomial of a graph is a 2-variable polynomial of significant im- portance in mathematics, statistical physics and biology [25]. In a strong sense it “contains” every graphical invariant that can be computed by deletion and contraction. -
The Chromatic Polynomial
Eotv¨ os¨ Lorand´ University Faculty of Science Master's Thesis The chromatic polynomial Author: Supervisor: Tam´asHubai L´aszl´oLov´aszPhD. MSc student in mathematics professor, Comp. Sci. Department [email protected] [email protected] 2009 Abstract After introducing the concept of the chromatic polynomial of a graph, we describe its basic properties and present a few examples. We continue with observing how the co- efficients and roots relate to the structure of the underlying graph, with emphasis on a theorem by Sokal bounding the complex roots based on the maximal degree. We also prove an improved version of this theorem. Finally we look at the Tutte polynomial, a generalization of the chromatic polynomial, and some of its applications. Acknowledgements I am very grateful to my supervisor, L´aszl´oLov´asz,for this interesting subject and his assistance whenever I needed. I would also like to say thanks to M´artonHorv´ath, who pointed out a number of mistakes in the first version of this thesis. Contents 0 Introduction 4 1 Preliminaries 5 1.1 Graph coloring . .5 1.2 The chromatic polynomial . .7 1.3 Deletion-contraction property . .7 1.4 Examples . .9 1.5 Constructions . 10 2 Algebraic properties 13 2.1 Coefficients . 13 2.2 Roots . 15 2.3 Substitutions . 18 3 Bounding complex roots 19 3.1 Motivation . 19 3.2 Preliminaries . 19 3.3 Proving the theorem . 20 4 Tutte's polynomial 23 4.1 Flow polynomial . 23 4.2 Reliability polynomial . 24 4.3 Statistical mechanics and the Potts model . 24 4.4 Jones polynomial of alternating knots . -
Computing Graph Polynomials on Graphs of Bounded Clique-Width
Computing Graph Polynomials on Graphs of Bounded Clique-Width J.A. Makowsky1, Udi Rotics2, Ilya Averbouch1, and Benny Godlin13 1 Department of Computer Science Technion{Israel Institute of Technology, Haifa, Israel [email protected] 2 School of Computer Science and Mathematics, Netanya Academic College, Netanya, Israel, [email protected] 3 IBM Research and Development Laboratory, Haifa, Israel Abstract. We discuss the complexity of computing various graph poly- nomials of graphs of fixed clique-width. We show that the chromatic polynomial, the matching polynomial and the two-variable interlace poly- nomial of a graph G of clique-width at most k with n vertices can be computed in time O(nf(k)), where f(k) ≤ 3 for the inerlace polynomial, f(k) ≤ 2k + 1 for the matching polynomial and f(k) ≤ 3 · 2k+2 for the chromatic polynomial. 1 Introduction In this paper1 we deal with the complexity of computing various graph poly- nomials of a simple graph of clique-width at most k. Our discussion focuses on the univariate characteristic polynomial P (G; λ), the matching polynomial m(G; λ), the chromatic polynomial χ(G; λ), the multivariate Tutte polyno- mial T (G; X; Y ) and the interlace polynomial q(G; XY ), cf. [Bol99], [GR01] and [ABS04b]. All these polynomials are not only of graph theoretic interest, but all of them have been motivated by or found applications to problems in chemistry, physics and biology. We give the necessary technical definitions in Section 3. Without restrictions on the graph, computing the characteristic polynomial is in P, whereas all the other polynomials are ]P hard to compute. -
Acyclic Orientations of Graphsଁ Richard P
Discrete Mathematics 306 (2006) 905–909 www.elsevier.com/locate/disc Acyclic orientations of graphsଁ Richard P. Stanley Department of Mathematics, University of California, Berkeley, Calif. 94720, USA Abstract Let G be a finite graph with p vertices and its chromatic polynomial. A combinatorial interpretation is given to the positive p p integer (−1) (−), where is a positive integer, in terms of acyclic orientations of G. In particular, (−1) (−1) is the number of acyclic orientations of G. An application is given to the enumeration of labeled acyclic digraphs. An algebra of full binomial type, in the sense of Doubilet–Rota–Stanley, is constructed which yields the generating functions which occur in the above context. © 1973 Published by Elsevier B.V. 1. The chromatic polynomial with negative arguments Let G be a finite graph, which we assume to be without loops or multiple edges. Let V = V (G) denote the set of vertices of G and X = X(G) the set of edges. An edge e ∈ X is thought of as an unordered pair {u, v} of two distinct vertices. The integers p and q denote the cardinalities of V and X, respectively. An orientation of G is an assignment of a direction to each edge {u, v}, denoted by u → v or v → u, as the case may be. An orientation of G is said to be acyclic if it has no directed cycles. Let () = (G, ) denote the chromatic polynomial of G evaluated at ∈ C.If is a non-negative integer, then () has the following rather unorthodox interpretation. Proposition 1.1. -
THE CHROMATIC POLYNOMIAL 1. Introduction a Common Problem in the Study of Graph Theory Is Coloring the Vertices of a Graph So Th
THE CHROMATIC POLYNOMIAL CODY FOUTS Abstract. It is shown how to compute the Chromatic Polynomial of a sim- ple graph utilizing bond lattices and the M¨obiusInversion Theorem, which requires the establishment of a refinement ordering on the bond lattice and an exploration of the Incidence Algebra on a partially ordered set. 1. Introduction A common problem in the study of Graph Theory is coloring the vertices of a graph so that any two connected by a common edge are different colors. The vertices of the graph in Figure 1 have been colored in the desired manner. This is called a Proper Coloring of the graph. Frequently, we are concerned with determining the least number of colors with which we can achieve a proper coloring on a graph. Furthermore, we want to count the possible number of different proper colorings on a graph with a given number of colors. We can calculate each of these values by using a special function that is associated with each graph, called the Chromatic Polynomial. For simple graphs, such as the one in Figure 1, the Chromatic Polynomial can be determined by examining the structure of the graph. For other graphs, it is very difficult to compute the function in this manner. However, there is a connection between partially ordered sets and graph theory that helps to simplify the process. Utilizing subgraphs, lattices, and a special theorem called the M¨obiusInversion Theorem, we determine an algorithm for calculating the Chromatic Polynomial for any graph we choose. Figure 1. A simple graph colored so that no two vertices con- nected by an edge are the same color. -
New Connections Between Polymatroids and Graph Coloring
New Connections between Polymatroids and Graph Coloring Joseph E. Bonin The George Washington University Joint work with Carolyn Chun These slides are available at http://blogs.gwu.edu/jbonin/ Polymatroids A (discrete or integer) polymatroid on a (finite) set E is a function E ρ : 2 ! Z that is I normalized: ρ(;) = 0, I non-decreasing: ρ(A) ≤ ρ(B) for all A ⊆ B ⊆ E, and I submodular: ρ(A [ B) + ρ(A \ B) ≤ ρ(A) + ρ(B) for all A; B ⊆ E. Polymatroids generalize matroids. Elements in polymatroids are loops, points, lines, planes, ::: A rank-4 polymatroid. a d c Any two lines except a; e and b; d are coplanar. b e f Point f is on line e: ρ(f ) = 1 & ρ(e) = ρ(e; f ) = 2. k-Polymatroids and Minors A polymatroid ρ on E is a k-polymatroid if ρ(e) ≤ k for all e 2 E. In a k-polymatroid, ρ(X ) ≤ k jX j for all X ⊆ E. Matroids are 1-polymatroids. Minors are defined as for matroids: for A ⊆ E, I deletion: ρnA(X ) = ρ(X ) for X ⊆ E − A, I contraction: ρ=A(X ) = ρ(X [ A) − ρ(A) for X ⊆ E − A, I minors: any combination of deletion and contraction. The k-dual The k-dual ρ∗ of a k-polymatroid ρ on E: for X ⊆ E, ρ∗(X ) = k jX j − ρ(E) + ρ(E − X ): ρ rank 4 ρ∗(E) = 6 Each element is a line in ρ∗. a d ρ∗(a; b) = ρ∗(d; e) = 3; other pairs have rank 4. -
Computation of Chromatic Polynomials Using Triangulations and Clique Trees
Computation of chromatic polynomials using triangulations and clique trees P. Berthom´e1, S. Lebresne2, and K. Nguy^en1 1 Laboratoire de Recherche en Informatique (LRI), CNRS UMRe 8623, Universit´e Paris-Sud, 91405, Orsay-Cedex, France, fPascal.Berthome,[email protected] 2 Preuves, Programmes et Syst`emes (PPS), CNRS UMR 7126, Universit´e Paris 7, Case 7014, 2 Place Jussieu, 75251 PARIS Cedex 05, France, [email protected] Abstract In this paper, we present a new algorithm for computing the chromatic polynomial of a general graph G. Our method is based on the addition of edges and contraction of non-edges of G, the base case of the recursion being chordal graphs. The set of edges to be considered is taken from a triangulation of G. To achieve our goal, we use the properties of triangulations and clique-trees with respect to the previous operations, and guide our algorithm to efficiently divide the original problem. Furthermore, we give some lower bounds of the general complexity of our method, and provide experimental results for several families of graphs. Finally, we exhibit an original measure of a triangulation of a graph. Keywords: Chromatic polynomial, chordal graphs, minimal triangula- tion, clique tree. R´esum´e Dans cet article, nous pr´esentons un nouvel algorithme pour calculer le polyn^ome chromatique d'un graphe quelconque G. Notre m´ethode s'appuie sur l'ajout r´ecursif d'ar^etes et de contraction de non-ar^etes de G, la r´ecursion s'arr^etant pour les graphes triangul´es. L'ensemble des ar^etes que l'on consid`ere dans ce processus forme une une triangulation de G. -
Various Numerical Invariants for Isolated Singularities Author(S): Stephen S.-T
Various Numerical Invariants for Isolated Singularities Author(s): Stephen S.-T. Yau Source: American Journal of Mathematics, Vol. 104, No. 5 (Oct., 1982), pp. 1063-1100 Published by: The Johns Hopkins University Press Stable URL: http://www.jstor.org/stable/2374084 . Accessed: 14/09/2011 23:46 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics. http://www.jstor.org VARIOUSNUMERICAL INVARIANTS FOR ISOLATEDSINGULARITIES By STEPHEN S.-T. YAu* 1. Introduction. In the theory of isolated singularities, one always wants to find invariants associated to the isolated singularities. Hopefully with enough invariants found, one can distinguish between isolated singu- larities. However, not many invariants are known. Definition 0. Let V be a Stein analytic space with x as its only singu- lar point. Let 7r:M -- V be a resolution of the singularity of V. We shall denote dim H1(M, 9), 1 < i < n - 1 by h(i), and dim Hq(M, UP) for 1 ? p < n, 1 < q < n by hP,q(M). So far as the classification problem is concerned, h (n-i) is one of the most important invariants. -
Graph Colouring: from Structure to Algorithms
Report from Dagstuhl Seminar 19271 Graph Colouring: from Structure to Algorithms Edited by Maria Chudnovsky1, Daniel Paulusma2, and Oliver Schaudt3 1 Princeton University, US, [email protected] 2 Durham University, GB, [email protected] 3 RWTH Aachen, DE, [email protected] Abstract This report documents the program and the outcomes of Dagstuhl Seminar 19271 “Graph Col- ouring: from Structure to Algorithm”, which was held from 30 June to 5 July 2019. The report contains abstracts for presentations about recent structural and algorithmic developments for the Graph Colouring problem and variants of it. It also contains a collection of open problems on graph colouring which were posed during the seminar. Seminar June 30–July 5, 2019 – http://www.dagstuhl.de/19271 2012 ACM Subject Classification Mathematics of computing → Graph coloring, Mathematics of computing → Graph algorithms Keywords and phrases (certifying / parameterized / polynomial-time) algorithms, computa- tional complexity, graph colouring, hereditary graph classes Digital Object Identifier 10.4230/DagRep.9.6.125 Edited in cooperation with Cemil Dibek 1 Executive Summary Maria Chudnovsky (Princeton University, US) Daniel Paulusma (Durham University, GB) Oliver Schaudt (RWTH Aachen, DE) License Creative Commons BY 3.0 Unported license © Maria Chudnovsky, Daniel Paulusma, and Oliver Schaudt The Graph Colouring problem is to label the vertices of a graph with the smallest possible number of colours in such a way that no two neighbouring vertices are identically coloured. Graph Colouring has been extensively studied in Computer Science and Mathematics due to its many application areas crossing disciplinary boundaries. Well-known applications of Graph Colouring include map colouring, job or timetable scheduling, register allocation, colliding data or traffic streams, frequency assignment and pattern matching.