Graphs of Polynomial Functions

Recognizing circulant graphs of prime order in polynomial time ∗ Mikhail E. Muzychuk Netanya Academic College 42365 Netanya, Israel mikhail@netvision.net.il Gottfried Tinhofer Technical University of Munich 80290 M¨unchen, Germany gottin@mathematik.tu-muenchen.de Submitted: December 19, 1997; Accepted: April 1, 1998 Abstract A circulant graph G of order n is a Cayley graph over the cyclic group Z n . Equivalently, G is circulant iff its vertices can be ordered such that the cor- responding adjacency matrix becomes a circulant matrix. To each circulant graph we may associate a coherent configuration A and, in particular, a Schur ring S isomorphic to A. A can be associated without knowing G to be circu- lant. If n is prime, then by investigating the structure of A either we are able to find an appropriate ordering of the vertices proving that G is circulant or we are able to prove that a certain necessary condition for G being circulant is violated. The algorithm we propose in this paper is a recognition algorithm for cyclic association schemes. It runs in time polynomial in n. MR Subject Number: 05C25, 05C85, 05E30 Keywords: Circulant graph, cyclic association scheme, recognition algorithm ∗ The work reported in this paper has been partially supported by the German Israel Foundation for Scientific Research and Development under contract # I-0333-263.06/93 the electronic journal of combinatorics 3 (1996), #Rxx 2 1 Introduction The graphs considered in this paper are of the form (X, γ), where X is a finite set and γ is a binary relation on X which is not necessarily symmetric. Let G be a group and G =(X, γ) a graph with vertex set X = G and with adjacency relation γ defined with the aid of some subset C ⊂Gby γ = {(g, h):g,h ∈G∧gh −1 ∈ C}. Then G is called Cayley graph over the group G. Let Z n , n ∈ N, stand for a cyclic group of order n written additively. A circulant graph G over Z n is a Cayley graph over this group. In this particular case, the adjacency relation γ has the form γ = n−1  i=0 {i}×{i+γ(0)} where γ(0) is the set of successors of the vertex 0. Evidently, the set of successors γ(i) of an arbitrary vertex i satisfies γ(i)=i+γ(0). The set γ(0) is called the connection set of the circulant graph G. G is a simple undirected graph if 0 ∈ γ(0) and j ∈ γ(0) implies −j ∈ γ(0). There are different equivalent characterizations of circulant graphs. One of them is this: A graph G is a circulant graph iff its vertex set can be numbered in such a way that the resulting adjacency matrix A(G) is a circulant matrix. We call such a numbering a Cayley numbering. Still another characterization is: G is a circulant graph iff a cyclic permutation of its vertices exists which is an automorphism of G. Cayley graphs, and in particular, circulant graphs have been studied intensively in the literature. These graphs are easily seen to be vertex transitive. In the case of a prime vertex number n circulant graphs are known to be the only vertex transitive graphs. Because of their high symmetry, Cayley graphs are ideal models for commu- nication networks. Routing and weight balancing is easily done on such graphs. Assume that a graph G on the set V (G)={0, ,n−1} is given by its diagram or by its adjacency matrix, or by some other data structure commonly used in dealing with graphs. How can we decide whether G is a Cayley graph or not? In such a generality, this decision problem seems to be far from beeing tractable efficiently. A recognition algorithm for Cayley graphs would have to involve implicitly checking all finite groups of order n. In the special case of circulant graphs, or in any other case where the group G is given, we could recognize Cayley graphs by checking all different numberings of the vertex set and comparing the corresponding adjacency matrix with the group table of G.Thisad hoc procedure is of course not efficient. the electronic journal of combinatorics 3 (1996), #Rxx 3 To our knowledge the first result towards recognizing circulant Graphs of Polynomial Functions Graphs of Polynomial Functions By: OpenStaxCollege The revenue in millions of dollars for a fictional cable company from 2006 through 2013 is shown in [link] Year 2006 2007 2008 2009 2010 2011 2012 2013 Revenues 52.4 52.8 51.2 49.5 48.6 48.6 48.7 47.1 The revenue can be modeled by the polynomial function R(t) = − 0.037t4 + 1.414t3 − 19.777t2 + 118.696t − 205.332 where R represents the revenue in millions of dollars and t represents the year, with t = corresponding to 2006 Over which intervals is the revenue for the company increasing? Over which intervals is the revenue for the company decreasing? These questions, along with many others, can be answered by examining the graph of the polynomial function We have already explored the local behavior of quadratics, a special case of polynomials In this section we will explore the local behavior of polynomials in general Recognizing Characteristics of Graphs of Polynomial Functions Polynomial functions of degree or more have graphs that not have sharp corners; recall that these types of graphs are called smooth curves Polynomial functions also display graphs that have no breaks Curves with no breaks are called continuous [link] shows a graph that represents a polynomial function and a graph that represents a function that is not a polynomial 1/46 Graphs of Polynomial Functions Recognizing Polynomial Functions Which of the graphs in [link] represents a polynomial function? 2/46 Graphs of Polynomial Functions The graphs of f and h are graphs of polynomial functions They are smooth and continuous 3/46 Graphs of Polynomial Functions The graphs of g and k are graphs of functions that are not polynomials The graph of function g has a sharp corner The graph of function k is not continuous Q&A Do all polynomial functions have as their domain all real numbers? Yes Any real number is a valid input for a polynomial function Using Factoring to Find Zeros of Polynomial Functions Recall that if f is a polynomial function, the values of x for which f(x) = are called zeros of f If the equation of the polynomial function can be factored, we can set each factor equal to zero and solve for the zeros We can use this method to find x-intercepts because at the x-intercepts we find the input values when the output value is zero For general polynomials, this can be a challenging prospect While quadratics can be solved using the relatively simple quadratic formula, the corresponding formulas for cubic and fourth-degree polynomials are not simple enough to remember, and formulas not exist for general higher-degree polynomials Consequently, we will limit ourselves to three cases in this section: The polynomial can be factored using known methods: greatest common factor and trinomial factoring The polynomial is given in factored form Technology is used to determine the intercepts How To Given a polynomial function f, find the x-intercepts by factoring Set f(x) = If the polynomial function is not given in factored form: Factor out any common monomial factors Factor any factorable binomials or trinomials Set each factor equal to zero and solve to find the x-intercepts Finding the x-Intercepts of a Polynomial Function by Factoring Find the x-intercepts of f(x) = x6 − 3x4 + 2x2 We can attempt to factor this polynomial to find solutions for f(x) = 4/46 Graphs of Polynomial Functions Factor out the greatest x6 − 3x4 + 2x2 = common factor x2(x4 − 3x2 + 2) = Factor the trinomial x2(x2 − 1)(x2 − 2) = Set each factor equal to zero (x2 − 1) = x2 = (x2 − 2) = x2 = or x=0 or x= ±1 x2 = x = ± √2 This gives us five x-intercepts: (0, 0), (1, 0), ( − 1, 0), (√2, 0), and ( − √2, 0) See [link] We can see that this is an even function Finding the x-Intercepts of a Polynomial Function by Factoring Find the x-intercepts of f(x) = x3 − 5x2 − x + Find solutions for f(x) = by factoring x3 − 5x2 − x + = x2(x − 5) − (x − 5) = (x2 − 1)(x − 5) = (x + 1)(x − 1)(x − 5) = x+1=0 x= −1 Factor by grouping Factor out the common factor Factor the difference of squares Set each factor equal to zero or x − = or x − = x=1 x=5 5/46 Graphs of Polynomial Functions There are three x-intercepts: ( − 1, 0), (1, 0), and (5, 0) See [link] Finding the y- and x-Intercepts of a Polynomial in Factored Form Find the y- and x-intercepts of g(x) = (x − 2)2(2x + 3) The y-intercept can be found by evaluating g(0) g(0) = (0 − 2)2(2(0) + 3) = 12 So the y-intercept is (0, 12) The x-intercepts can be found by solving g(x) = (x − 2)2(2x + 3) = (x − 2)2 = x−2=0 (2x + 3) = or x= − x=2 6/46 Graphs of Polynomial Functions ( ) So the x-intercepts are (2, 0) and − , Analysis We can always check that our answers are reasonable by using a graphing calculator to graph the polynomial as shown in [link] Finding the x-Intercepts of a Polynomial Function Using a Graph Find the x-intercepts of h(x) = x3 + 4x2 + x − 7/46 Graphs of Polynomial Functions This polynomial is not in factored form, has no common factors, and does not appear ...Recognizing circulant graphs in polynomial time: An application of association schemes Mikhail E. Muzychuk ∗ , Department of Computer Science and Mathematics, Netanya Academic College, Netanya, 42365, Israel muzy@netanya.ac.il Gottfried Tinhofer, Zentrum Mathematik, Technical University of Munich, 80290 Munich, Germany gottin@mathematik.tu-muenchen.de Submitted: October 7, 2000; Accepted: May 26, 2001 MR subject classifications: 05C25, 05C85. Abstract In this paper we present a time-polynomial recognition algorithm for certain classes of circulant graphs. Our approach uses coherent configurations and Schur rings generated by circulant graphs for elucidating their symmetry properties and eventually finding a cyclic automorphism. Key words: Circulant graphs, association schemes, Schur rings. 1 Introduction We consider graphs of the form G =(X, γ), where X is a finite set and γ is a binary relation on X,theadjacency relation.Forx ∈ X put γ(x)={y :(x, y) ∈ γ}. ∗ Partially supported by DAAD fellowship A/00/24054 the electronic journal of combinatorics 8 (2001), #R26 1 Let G be a group and G =(X, γ) a graph with vertex set X = G and with adjacency relation γ defined with the aid of some subset S ⊂ G by γ = {(g, h):g,h ∈ G ∧ hg −1 ∈ S}. Then G is called Cayley graph over the group G and S is called connection set of G. Let Z n , n ∈ N, stand for a cyclic group of order n, written additively. A circulant graph G of order n (or a circulant, for short) is a Cayley graph over Z n . In this particular case, the adjacency relation γ has the form γ = n−1  i=0 {i}×{i + γ(0)} where γ(0) is the set of successors of the vertex 0. Evidently, the set of successors γ(i)of an arbitrary vertex i satisfies γ(i)=i + γ(0). All arithmetic operations with vertex num- bers are understood modulo n. We do not distinguish by notation between the element z ∈ Z n and the integer z ∈ Z. From the context, it will always be clear what is meant. For a ∈ Z n and S ⊂ Z n we write aS for the set {as | s ∈ S} . For a circulant G the connection set is γ(0). G is a simple undirected graph if 0 ∈ γ(0) and if j ∈ γ(0) implies −j ∈ γ(0). There are different equivalent characterizations of circulants. One of them is this: A graph G is a circulant iff its vertex set can be numbered in such a way that the resulting adjacency matrix A(G) is a circulant matrix. We call such a numbering a Cayley num- bering. Still another characterization is: G is a circulant iff a cyclic permutation of its vertices exists which is an automorphism of G. Such an automorphism we shall call a full cycle. Cayley graphs, and in particular circulants, have been studied intensively in the lit- erature. These graphs are vertex-transitive. In the case of a prime vertex number n, circulants are known to be the only vertex-transitive graphs. Because of their high sym- metry, Cayley graphs are ideal models for communication networks. In this context, recently particular interest has been awaken for so-called geometric circulants. A ge- ometric circulant GC(n, d) is a circulant on the vertex set Z n possessing a connection set γ(0) = {±1, ±d, ±d 2 , ,±d m }, consisting of a geometric progression in d and its inverses, where d is a natural number satisfying 1 <d≤ n 2 and m is such that d m +1<n≤ d m+1 +1. Certain geometric circulant graphs have been proposed in [22] as a new topology for multicomputer networks. The circulants in this paper have been called recursive circu- lants, they are geometric circulants with vertex number n = cd m for some c, 1 <c≤ d. the electronic journal of combinatorics 8 (2001), #R26 2 The motivation for the attribute recursive, as pointed out in this paper, is the fact that circulants GC(cd m ,d) possess a hierarchical structure. If one drops all edges in GC(cd m ,d) which are of the form (v, v ± 1) then the remaining graph is a union of d graphs, each isomorphic to GC(cd m−1 ,d). A hierarchy like this, however, may be observed also in more general situations. Cayley graphs A multivariate interlace polynomial and its computation for graphs of bounded clique-width Bruno Courcelle ∗ Institut Universitaire de France and Bordeaux University, LaBRI courcell@labri.fr Submitted: Jul 31, 2007; Accepted: Apr 30, 2008; Published: May 5, 2008 Mathematics Subject Classifications: 05A15, 03C13 Abstract We define a multivariate polynomial that generalizes in a unified way the two- variable interlace polynomial defined by Arratia, Bollob´as and Sorkin on the one hand, and a one-variable variant of it defined by Aigner and van der Holst on the other. We determine a recursive definition for our polynomial that is based on local complementation and pivoting like the recursive definitions of Tutte’s polynomial and of its multivariate generalizations are based on edge deletions and contractions. We also show that bounded portions of our polynomial can be evaluated in polyno- mial time for graphs of bounded clique-width. Our proof uses an expression of the interlace polynomial in monadic second-order logic, and works actually for every polynomial expressed in monadic second-order logic in a similar way. 1 Introduction There exist a large variety of polynomials associated with graphs, matroids and combi- natorial maps. They provide information about configurations in these objects. We take here the word “configuration” in a wide sense. Typical examples are colorings, matchings, stable subsets, subgraphs. In many cases, a value is associated with the considered config- urations : number of colors, cardinality, number of connected components or rank of the adjacency matrix of an associated subgraph. The information captured by a polynomial can be recovered in three ways: either by evaluating the polynomial for specific values of the indeterminates, or from its zeros, or by interpreting the coefficients of its monomials. We will consider the latter way in this article. ∗ This work has been supported by the GRAAL project of “Agence Nationale pour la Recherche” and by a temporary position of CNRS researcher. Postal address: LaBRI, F-33405 Talence, France the electronic journal of combinatorics 15 (2008), #R69 1 A multivariate polynomial is a polynomial with indeterminates depending on the ver- tices or the edges of the considered graph. Such indeterminates are sometimes called colors or weights because they make it possible to evaluate the polynomial with distinct values associated with distinct vertices or edges. Several multivariate versions of the dichromatic and Tutte polynomials of a graph have been defined and studied by Traldi in [30], by Za- slavsky in [ 31], by Bollob´as and Riordan in [6] and by Ellis-Monaghan and Traldi who generalize and unify in [16] the previous definitions. Motivated by problems of statistical physics, Sokal studies in [29] a polynomial that will illustrate this informal presentation. The multivariate Tutte polynomial of a graph G = (V, E) is defined there as: Z(G) =  A⊆E u k(G[A])  e∈A v e where G[A] is the subgraph of G with vertex set V and edge set A, and k(G[A]) is the number of its connected components. This polynomial belongs to Z[u, v e ; e ∈ E]. An indeterminate v e is associated with each edge e. The indeterminates commute, the order of enumeration over each set A is irrelevant. We call such an expression an explicit definition of Z(G), to be contrasted with its recursive definition, formulated as follows ([29], Formula (4.16)) in terms of edge deletions and contractions: Z(G) = u |V | if G has no edge, Z(G) = Z(G[E − {e}]) + v e · Z(G/e) if e is any edge, where G/e is obtained from G by contracting edge e. From the fact that Z(G) satisfies these equalities, it follows that they form a recursive definition which is well-defined in the sense that it yields the same result for every choice of an edge e in the second clause, i.e., for every tree of recursive calls. There is no general method for constructing a recursive definition from an explicit one or proving that such a definition does not exist. The Evaluating a Weighted Graph Polynomial for Graphs of Bounded Tree-Width S. D. Noble Department of Mathematical Sciences Brunel University Kingston Lane, Uxbridge UB8 3PH, U.K. steven.noble@brunel.ac.uk Submitted: Oct 30, 2006; Accepted: May 18, 2009; Published: May 29, 2009 Mathematics S ubject Classification: 05C85, 05C15, 68R10 Abstract We show that for any k there is a polynomial time algorithm to evaluate the weighted graph p olynomial U of any graph with tree-width at most k at any point. For a graph with n vertices, the algorithm requires O(a k n 2k+ 3 ) arithm etical opera- tions, where a k depends only on k. 1 Introduction Motivated by a series of papers [9, 10, 1 1], the weighted graph polynomial U was intro- duced in [22]. Chmutov, Duzhin and Lando [9, 10, 11] introduce a graph polynomial derived from Vassiliev invariants of knots and note that this polynomial does not include the Tutte polynomial as a special case. With a slight generalisation of their definition we obtain the weighted graph polynomial U that does include the Tutte polynomial. The attraction of U is that it contains many other graph invariants as specialisations, for instance the 2 -polymatroid rank generating function of O xley and Whittle [23], and as a consequence the matching polynomial, the stable set polynomial [13] and the symmetric function generalisation of the chromatic polynomial [27]. Note however that there are non-isomorphic graphs with the same U polynomial. This is a corolla ry of a result of Sarmiento [26], showing that the coefficients of U and the polychromate determine one another. It remains an open problem to determine whether or not there are two non- isomorphic trees with the same U polynomial. We introduce U in Section 2 and review some of these results in more detail. The notion of tree-width was introduced by Robertson and Seymour as a key tool in their work on the graph minors project [24, 25]. An equivalent notion, studied extensively by Arnborg and Proskurowski, (see for instance [3, 4]), is that of a partial k-tree. the electronic journal of combinatorics 16 (2009), #R64 1 Many well-studied classes of graphs have bo unded tree-width: for instance, series- parallel networks are the graphs with tree-width at most two. A large class of g r aph problems, which are thought to be intractable, can be solved when the input is restricted to graphs with tree-width at most a fixed constant k. For example, the NP-complete problems, 3-Colouring and Hamiltonian Circuit can be solved in linear time for graphs of bounded tree-width [4]. For a g ood survey of tree-width see [5]. When the underlying graph is obvious, we let n be its number of vertices, m be its number of edges and p be the largest size of a set of mutually parallel edges. Theorem 1.1. For any k ∈ N, there exists an algorithm A k with the following prop- erties. The input is any graph G, with tree-width at most k, and rationals x 1 = p 1 /q 1 , . . . , x n = p n /q n and y = p 0 /q 0 such that for all i, p i and q i are coprime. The output is U G (x 1 , . . . , x n , y); the running time is O(a k n 2k+ 3 (n 2 + m)r log p log(r(n + m)) log(log(r(n + m)))), where r = log(max{|p 0 |, . . . , |p n |, |q 0 |, . . . , |q n |}) and a k depends only on k. The result extends that of [2 0] and independently [2] where an algorithm to evaluate the Tutte p olynomial of a graph having tree-width at most k is presented. In [20], the algorithm given requires only a linear (in n) number of multiplications. Despite using the same basic idea as in [20], we are unable to reduce the amount of computational effort required to evaluate U down to O(n α ) operations, where α is independent of k. More recently Hlinˇen´y [15] has shown that the Tutte polynomial is computable in polynomial time when the input is restricted to matroids with bounded branchwidth rep- resentable over a finite field. Furthermore Makowsky [17] and Makowsky and Mari˜no [19] have shown that there are polynomial time alg orithms to LOCAL POLYNOMIAL CONVEXITY OF GRAPHS OF FUNCTIONS IN SEVERAL VARIABLES KIEU PHUONG CHI Abstract. In that paper, we investigate the locally polynomial convexity of graphs of smooth functions in several variables. We also give a sufficient condition for real analytic function g defined near 0 in C which behaves like z n near the origin so that the algebra generated by z m and g is dense in the space of continuous functions on D for all disks D close enough to the origin in C. 1. Introduction ˆ we denote the polynomial convex We recall that for a given compact K in Cn , by K hull of K i.e., ˆ = {z ∈ Cn : |p(z)| ≤ p K K for every polynomial p in Cn }. ˆ = K. A compact K is called locally We say that K is polynomially convex if K polynomially convex at a ∈ K if there exists the closed ball B(a) centered at a such that B(a) ∩ K is polynomially convex. The interest for studying polynomial convexity stems from the celebrated Oka-Weil approximation theorem (see [1], page 36) which states that holomorphic functions near a compact polynomially convex subset of Cn can be uniformly approximated by polynomials in Cn . A compact K ⊂ C is polynomially convex if is C \ K connected. In higher dimensions, there is no such topological characterization of polynomially convex sets, and it is usual difficult to determine whether a given compact subset is polynomially convex. By a well-known result of Wermer ([19]; see also [1], Theorem. 17.1), every totally real manifold is locally polynomially convex. Recall that a C 1 smooth real manifold M is called totally real at p ∈ M if the real tangent space Tp M contains no complex line. In this paper, we are concerned with local polynomial convexity at the origin of the graph Γf of a C 2 smooth function f near 0 ∈ Cn such that f (0) = 0. By the theorem of Wermer just cited, we ∂f know that if (0) = 0 for all i = 1, 2, ..., n then Γf is locally polynomially convex at ∂z i ∂f the origin of Cn+1 . Thus it remains to consider the case where (0) = 0 for some i. ∂z i Our study is motivated by a similar problem in one complex variable. More precisely, let f be a C 2 smooth function near 0 ∈ C such that f (0) = 0. Under certain condition of f , one can show that Γf is locally polynomial convex at the origin of C2 . The work 2010 Mathematics Subject Classification. 46J10, 46J15, 47H10. Key words and phrases. polynomially convex, plurisubharmonic, totally real . 1 associated with these direction of research is too numerous to list here; instead, the reader is referred to [2, 3, 20] and the references given therein. In the section 3, we will refine the technical from [6] to attack the problem in several variables. For the readers convenience, we repeat a reasoning due to [6]. First, we construct nonnegative smooth functions vanishing exactly on Γf . These functions are, in general, plurisubharmonic only on open sets whose boundaries contain the origin. Secondly,under some technical assumptions, we may add small strictly plurisubharmonic functions to obtain plurisubharmonic functions on certain open sets containing the (local) polynomially convex hull of Γf . Finally, by invoking the nontrivial fact of about equivalence of plurisubharmonic hull and polynomial hulls, we can conclude that Γf is locally polynomially convex at the origin. In this vein, we obtain some known results in one variable. We also give some examples to show that our results are effective. In section 4, we shall present some results about locally uniform approximation of continuous function. Let D be a small closed disk in the complex plane, centered at the origin and g be a C 2 function on D which behaves like z n near the origin. By [z m , g; D] we denote the function algebra consisting of uniform limits on D of all polynomials in z m and g. Our goal finding conditions on g .. .Graphs of Polynomial Functions Recognizing Polynomial Functions Which of the graphs in [link] represents a polynomial function? 2/46 Graphs of Polynomial Functions The graphs of f and h are graphs. .. are graphs of polynomial functions They are smooth and continuous 3/46 Graphs of Polynomial Functions The graphs of g and k are graphs of functions that are not polynomials The graph of function... negative, it will change the direction of the end behavior [link] summarizes all four cases 14/46 Graphs of Polynomial Functions 15/46 Graphs of Polynomial Functions Understanding the Relationship