DOCSLIB.ORG

A Graph Polynomial Approach to Primitivity⋆

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Graph Polynomial Approach to Primitivity⋆ A Graph Polynomial Approach to Primitivity? F. Blanchet-Sadri1, Michelle Bodnar2, Nathan Fox3, and Joe Hidakatsu2 1 Department of Computer Science, University of North Carolina, P.O. Box 26170, Greensboro, NC 27402{6170, USA 2 Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109{1043, USA 3 Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd., Piscataway, NJ 08854{8019, USA Abstract. Recently, Tittmann et al. introduced the subgraph component polynomial and showed that its power for distinguishing graphs is quite different from the power of other graph polynomials that appear in the literature such as the matching polynomial, the Tutte polynomial, the characteristic polynomial, the chromatic polynomial, etc. The subgraph component polynomial enumerates vertex induced subgraphs in a given undirected graph with respect to the number of components. We show the use of the subgraph component polynomial to count the number of primitive partial words of a given length over an alphabet of a fixed size, which leads to a method for enumerating such partial words. 1 Introduction Motivated by social and biological networks, Tittmann et al. [10] introduced the subgraph component polynomial Q(G; x; y) of an undirected graph G with n vertices as the bivariate generating function which counts the number of con- nected components in vertex induced subgraphs. More precisely, Q(G; x; y) = n n i j Σi=0Σj=0qij(G)x y , where qij(G) is the number of vertex induced subgraphs of G with exactly i vertices and j connected components. They related the subgraph component polynomial to other graph polynomials that appear in the literature such as the Tutte polynomial, the universal edge elimination polynomial, etc. (see, for instance, [8] for more information on graph polynomials). They showed several remarkable properties of the subgraph component polynomial, among them is their use to compute the so-called \residual connectedness reliability". They also showed that the problem of computing Q(G; x; y) is ]P -hard, but that it is fixed parameter tractable when restricting to graph classes that have bounded tree-width and to classes of bounded clique-width. Primitivity is a well-studied topic in combinatorics on words (see, for in- stance, [4]). It is well-known that the number of primitive words of a given length over an alphabet of a fixed size can be calculated using the M¨obiusfunc- tion [7, 9]. In this paper, we discuss the use of the above subgraph component ? This material is based upon work supported by the National Science Foundation under Grant No. DMS{1060775. 2 F. Blanchet-Sadri, M. Bodnar, N. Fox, and J. Hidakatsu polynomial to count the number Ph;k(n) of primitive partial words with h holes of length n over a k-letter alphabet. Research on primitive partial words was initiated by the first author in [1]. Partial words, also referred to as strings with don't-cares, may have some undefined positions or holes. In [2], formulas for h = 1 and h = 2 are given in terms of the formula for h = 0 and some bounds are provided for h > 2, but no exact formulas are given for h > 2. Here, we associate a graph Gn;P with any partial word of length n with period set P as follows: the vertices represent the positions 0; : : : ; n − 1 and the edges are the pairs fi; i + mpg, where 0 ≤ i < i + mp ≤ n − 1, m 2 Z, and p 2 P. It turns out that Ph;k(n) can be expressed in terms of the Q(Gn;P ; x; y)'s. The contents of our paper are as follows: In Section 2, we answer the ques- tion \How many holes can a primitive partial word of length n over a k-letter alphabet contain?". We show that this number can be expressed in terms of the large factors of n. (Note that this count has recently led to efficient algo- rithms for computing all primitively-rooted squares and runs in partial words [3].) In Section 3, we describe a general method for counting primitive partial words with the subgraph component polynomial, which leads to a method for the enumeration of primitive partial words. In Section 4, we discuss in particular non-primitive partial words of length pq, where p and q are distinct primes. In doing so, we give a framework for understanding partial words that are exactly p- periodic and exactly q-periodic without being 1-periodic (this relates to a variant of Fine and Wilf's periodicity theorem [5]). In Section 5, we further discuss the n n−h computation of Nh;k(n) = h k − Ph;k(n), the number of non-primitive par- tial words with h holes of length n over a k-letter alphabet, using the subgraph component polynomial. Finally in Section 6, we conclude with some remarks. We end this section by reviewing a few basic concepts on partial words. Let A be a non-empty finite set, or an alphabet. We consider a partial word w over A as a word over the enlarged alphabet A = A [ {}, where the additional character plays the role of an undefined position or a hole. For 0 ≤ i < n, the character at position i of w is denoted by w(i). If w(i) 2 A, then i is defined, otherwise i is a hole. A full word is a partial word with an empty set of holes. We denote by w[i::j) the factor of w that starts at position i and ends at position j − 1, and by jwj the length of w or the number of characters in w. If w1 and w2 are two partial words of equal length, then w1 is contained in w2, denoted by w1 ⊂ w2, if w1(i) = w2(i) for all defined positions i in w1. The greatest lower bound of w1 and w2, denoted by w1 ^ w2, is the maximal partial word contained in both w1 and w2, i.e, (w1 ^w2) ⊂ w1 and (w1 ^w2) ⊂ w2, and if w ⊂ w1 and w ⊂ w2, then w ⊂ (w1^w2). For example, abba^aaa = a. For a positive integer p, a partial word w has a period of p or w is p-periodic if for all positions i; j defined in w such that i ≡ j mod p, we have w(i) = w(j). A partial word has an exact period of p or is exactly p-periodic if it is p-periodic and p divides its length. A partial word w is primitive if there exists no full word v such that w ⊂ vi with i ≥ 2, equivalently, if there is no proper factor p of jwj such that w is p-periodic. Clearly, if w is primitive and w ⊂ w0, then w0 is primitive. A Graph Polynomial Approach to Primitivity 3 2 Maximizing Number of Holes in Primitive Words We define the set of large factors LF (n) of an integer n as the set of integers m such that m < n, m j n, and for t 6= m; t j n, we have m - t. For example, when n = 30 we have LF (30) = f6; 10; 15g. Clearly, the large factors of n come from dividing n by its prime factors. Proposition 1. Given a primitive partial word of length n which contains the maximum number of holes, set LF (n) = ff1; : : : ; fmg. For any non-hole posi- tions i and j, we have i − j = c1f1 + ··· + cmfm for some ci 2 Z. Moreover, the fewest number of non-holes which rule out all of the large factors of n as periods is jLF (n)j + 1. Proof. To rule out the large factor f1, we must use two non-holes i1; i2 and they must differ by a multiple of f1. Note that for each i and j, we have lcm(fi; fj) = n, so i1 and i2 rule out at most one large factor, i.e., f1. To rule out the next large factor f2, there must exist a non-hole position i3 that differs from i1 or i2, say i1, by some multiple of f2. Since i1 − i2 = c1f1 and i1 − i3 = c2f2 for some c1; c2 2 Z, we get i2 − i3 = i2 − i1 + i1 − i3 = −c1f1 + c2f2. As noted earlier, the addition of i3 cannot possibly rule out any large factor other than f2. Continue in this way until all large factors have been ruled out as periods. After ruling out the first large factor with two non-holes, jLF (n)j − 1 non-holes are required to rule out the remaining jLF (n)j − 1 large factors. This necessitates a total of jLF (n)j + 1 non-hole positions to rule out all large factors of n. ut Theorem 1. The maximum number of holes that a primitive partial word of length n over an arbitrary alphabet of at least two letters can contain, denoted τ(n), is τ(n) = n − jLF (n)j − 1. Moreover, the maximum number of holes a primitive word can contain can be achieved using a binary alphabet. Proof. We begin with a construction over the binary alphabet fa; bg which shows that this number of holes can always be achieved. Let the word w be defined as w(i) = a if i + 1 2 LF (n), w(i) = b if i = n − 1, and w(i) = otherwise. There are jLF (n)j a's and one b, leaving room for exactly n−(jLF (n)j+1) holes as desired. We observe that it is unnecessary to check for incompatibilities in smaller periods because for any factor q which divides an element p of LF (n), an incompatibility in a period of length p implies an incompatibility in a period of length q, so we need only check that all periods given by our large factor set do not occur in w.
Load more

Recommended publications
  • THE N-DIMENSIONAL MATCHING POLYNOMIAL B. Lass
    GAFA, Geom. funct. anal. Vol. 15 (2005) 453 – 475 c Birkh¨auser Verlag, Basel 2005 1016-443X/05/020453-23 DOI 10.1007/s00039-005-0512-0 GAFA Geometric And Functional Analysis THE N-DIMENSIONAL MATCHING POLYNOMIAL B. Lass To P. Cartier and A.K. Zvonkin Abstract. Heilmann et Lieb ont introduit le polynˆome de couplage µ(G, x) d’un graphe G =(V,E). Nous prolongeons leur d´efinition en munissant chaque sommet de G d’une forme lin´eaire N-dimensionnelle (ou bien d’un vecteur) et chaque arˆete d’une forme sym´etrique bilin´eaire. On attache doncatout ` r-couplage de G le produit des formes lin´eaires des sommets qui ne sont pas satur´es par le couplage, multipli´e par le produit des poids des r arˆetes du couplage, o`u le poids d’une arˆete est la valeur de sa forme ´evalu´ee sur les deux vecteurs de ses extr´emit´es. En multipliant par (−1)r et en sommant sur tous les couplages, nous obtenons notre polynˆome de couplage N-dimensionnel. Si N =1,leth´eor`eme principal de l’article de Heilmann et Lieb affirme que tous les z´eros de µ(G, x)sontr´eels. Si N =2, cependant, nous avons trouv´e des graphes exceptionnels o`u il n’y a aucun z´ero r´eel, mˆeme si chaque arˆete est munie du produit scalaire canonique. Toutefois, la th´eorie de la dualit´ed´evelopp´ee dans [La1] reste valable en N dimensions. Elle donne notamment une nouvelle interpr´etationala ` transformation de Bargmann–Segal, aux diagrammes de Feynman et aux produits de Wick.
    [Show full text]
  • On Connectedness and Graph Polynomials
    ON CONNECTEDNESS AND GRAPH POLYNOMIALS by Lucas Mol Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at Dalhousie University Halifax, Nova Scotia April 2016 ⃝c Copyright by Lucas Mol, 2016 Table of Contents List of Tables ................................... iv List of Figures .................................. v Abstract ...................................... vii List of Abbreviations and Symbols Used .................. viii Acknowledgements ............................... x Chapter 1 Introduction .......................... 1 1.1 Background . .6 Chapter 2 All-Terminal Reliability ................... 11 2.1 An Upper Bound on the Modulus of any All-Terminal Reliability Root.................................... 13 2.1.1 All-Terminal Reliability and Simplical Complexes . 13 2.1.2 The Chip-Firing Game and Order Ideals of Monomials . 16 2.1.3 An Upper Bound on the Modulus of any All-Terminal Reliabil- ity Root . 19 2.2 All-Terminal Reliability Roots outside of the Unit Disk . 25 2.2.1 All-Terminal Reliability Roots of Larger Modulus . 25 2.2.2 Simple Graphs with All-Terminal Reliability Roots outside of the Unit Disk . 29 Chapter 3 Node Reliability ........................ 45 3.1 Monotonicity . 51 3.2 Concavity and Inflection Points . 66 3.3 Fixed Points . 74 3.4 The Roots of Node Reliability . 80 ii Chapter 4 The Connected Set Polynomial ............... 84 4.1 Complexity . 88 4.2 Roots of the Connected Set Polynomial . 99 4.2.1 Realness and Connected Set Roots . 99 4.2.2 Bounding the Connected Set Roots . 104 4.2.3 The Closure of the Collection of Connected Set Roots . 116 Chapter 5 The Subtree Polynomial ................... 130 5.1 Connected Sets and Convexity . 132 5.2 Paths and Stars .
    [Show full text]
  • Harary Polynomials
    numerative ombinatorics A pplications Enumerative Combinatorics and Applications ECA 1:2 (2021) Article #S2R13 ecajournal.haifa.ac.il Harary Polynomials Orli Herscovici∗, Johann A. Makowskyz and Vsevolod Rakitay ∗Department of Mathematics, University of Haifa, Israel Email: [email protected] zDepartment of Computer Science, Technion{Israel Institute of Technology, Haifa, Israel Email: [email protected] yDepartment of Mathematics, Technion{Israel Institute of Technology, Haifa, Israel Email: [email protected] Received: November 13, 2020, Accepted: February 7, 2021, Published: February 19, 2021 The authors: Released under the CC BY-ND license (International 4.0) Abstract: Given a graph property P, F. Harary introduced in 1985 P-colorings, graph colorings where each color class induces a graph in P. Let χP (G; k) counts the number of P-colorings of G with at most k colors. It turns out that χP (G; k) is a polynomial in Z[k] for each graph G. Graph polynomials of this form are called Harary polynomials. In this paper we investigate properties of Harary polynomials and compare them with properties of the classical chromatic polynomial χ(G; k). We show that the characteristic and the Laplacian polynomial, the matching, the independence and the domination polynomials are not Harary polynomials. We show that for various notions of sparse, non-trivial properties P, the polynomial χP (G; k) is, in contrast to χ(G; k), not a chromatic, and even not an edge elimination invariant. Finally, we study whether the Harary polynomials are definable in monadic second-order Logic. Keywords: Generalized colorings; Graph polynomials; Courcelle's Theorem 2020 Mathematics Subject Classification: 05; 05C30; 05C31 1.
    [Show full text]
  • The Complexity of Multivariate Matching Polynomials
    The Complexity of Multivariate Matching Polynomials Ilia Averbouch∗ and J.A.Makowskyy Faculty of Computer Science Israel Institute of Technology Haifa, Israel failia,[email protected] February 28, 2007 Abstract We study various versions of the univariate and multivariate matching and rook polynomials. We show that there is most general multivariate matching polynomial, which is, up the some simple substitutions and multiplication with a prefactor, the original multivariate matching polynomial introduced by C. Heilmann and E. Lieb. We follow here a line of investigation which was very successfully pursued over the years by, among others, W. Tutte, B. Bollobas and O. Riordan, and A. Sokal in studying the chromatic and the Tutte polynomial. We show here that evaluating these polynomials over the reals is ]P-hard for all points in Rk but possibly for an exception set which is semi-algebraic and of dimension strictly less than k. This result is analoguous to the characterization due to F. Jaeger, D. Vertigan and D. Welsh (1990) of the points where the Tutte polynomial is hard to evaluate. Our proof, however, builds mainly on the work by M. Dyer and C. Greenhill (2000). 1 Introduction In this paper we study generalizations of the matching and rook polynomials and their complexity. The matching polynomial was originally introduced in [5] as a multivariate polynomial. Some of its general properties, in particular the so called half-plane property, were studied recently in [2]. We follow here a line of investigation which was very successfully pursued over the years by, among others, W. Tutte [15], B.
    [Show full text]
  • Intriguing Graph Polynomials Intriguing Graph Polynomials
    ICLA-09, January 2009 Intriguing Graph Polynomials Intriguing Graph Polynomials Johann A. Makowsky Faculty of Computer Science, Technion - Israel Institute of Technology, Haifa, Israel http://www.cs.technion.ac.il/∼janos e-mail: [email protected] ********* Joint work with I. Averbouch, M. Bl¨aser, H. Dell, B. Godlin, T. Kotek and B. Zilber Reporting also recent work by M. Freedman, L. Lov´asz, A. Schrijver and B. Szegedy Graph polynomial project: http://www.cs.technion.ac.il/∼janos/RESEARCH/gp-homepage.html 1 ICLA-09, January 2009 Intriguing Graph Polynomials Overview • Parametrized numeric graph invariants and graph polynomials • Evaluations of graph polynomials • What we find intriguing • Numeric graph invariants: Properties and guiding examples • Connection matrices • MSOL-definable graph polynomials • Finite rank of connection matrices • Applications of the Finite Rank Theorem • Complexity of evaluations of graph polynomials • Towards a dichotomy theorem 2 ICLA-09, January 2009 Intriguing Graph Polynomials References, I [CMR] B. Courcelle, J.A. Makowsky and U. Rotics: On the Fixed Parameter Complexity of Graph Enumeration Problems Definable in Monadic Second Order Logic, Discrete Applied Mathematics, 108.1-2 (2001) 23-52 [M] J.A. Makowsky: Algorithmic uses of the Feferman-Vaught theorem, Annals of Pure and Applied Logic, 126.1-3 (2004) 159-213 [M-zoo] J.A. Makowsky: From a Zoo to a Zoology: Towards a general theory of graph polynomials, Theory of Computing Systems, Special issue of CiE06, online first, October 2007 3 ICLA-09, January 2009 Intriguing Graph Polynomials References, II [MZ] J.A. Makowsky and B. Zilber, Polynomial invariants of graphs and totally categorical theories, MODNET Preprint No.
    [Show full text]
  • Generalizations of the Matching Polynomial to the Multivariate
    Generalizations of the Matching Polynomial to the Multivariate Independence Polynomial Jonathan Leake Nick Ryder ∗ February 20, 2019 Abstract We generalize two main theorems of matching polynomials of undirected simple graphs, namely, real-rootedness and the Heilmann-Lieb root bound. Viewing the matching polynomial of a graph G as the independence polynomial of the line graph of G, we determine conditions for the extension of these theorems to the independence polynomial of any graph. In particular, we show that a stability-like property of the multivariate independence polynomial characterizes claw-freeness. Finally, we give and extend multivariate versions of Godsil’s theorems on the divisibility of matching polynomials of trees related to G. 1 Introduction Given a graph G = (V, E), the matching polynomial of G and the independence polynomial of G are defined as follows. |M| µ(G) := x2 I(G) := x|S| − MX⊂E SX⊂V M,matching S,independent The real-rootedness of the matching polynomial and the Heilmann-Lieb root bound are important results in the theory of undirected simple graphs. In particular, real-rootedness implies log-concavity and unimodality of the matchings of a graph, and recently in [MSS15] the root bound was used to show the existence of Ramanujan graphs. Additionally, it is well-known that the matching polynomial of a graph G is equal to the independence polynomial of the line graph of G. With this, one obtains the same results for the independence polynomials of line graphs. This then leads to a natural question: what properties extend to the independence polynomials of all graphs? Generalization of these results to the independence polynomial has been partially successful.
    [Show full text]
  • CHRISTOFFEL-DARBOUX TYPE IDENTITIES for INDEPENDENCE POLYNOMIAL3 Conjecture Turned out to Be False for General Graphs, As It Was Pointed out in [3]
    CHRISTOFFEL-DARBOUX TYPE IDENTITIES FOR INDEPENDENCE POLYNOMIAL FERENC BENCS Abstract. In this paper we introduce some Christoffel-Darboux type identities for independence polynomials. As an application, we give a new proof of a theorem of M. Chudnovsky and P. Seymour, claiming that the independence polynomial of a claw-free graph has only real roots. Another application is related to a conjecture of Merrifield and Simmons. 1. Introduction The independence polynomial of a graph G is defined by: I(G, x)= a (G)xk, X k k=0 where a0(G) = 1, and if k ≥ 1, then ak(G) denotes the number of independent sets of G of size k. The matching polynomial of a graph G is defined in a similar way: µ(G, x)= (−1)km (G)xn−2k, X k k=0 where m0(G) = 1, and if k ≥ 1, then mk(G) is the number of matchings of size k. The Christoffel-Darboux identity is one of the most important tools in the theory of orthogonal polynomials. It asserts, that if (pn(x)) is a sequence of orthogonal polynomials, then n 1 kn pn(y)pn+1(x) − pn(x)pn+1(x) pj(x)pj(y)= , X hj hnhn x − y j=0 +1 where hj is the squared norm of pj(x), and kj is the leading coefficient of pj(x). A one-line consequence of this identity is the real-rootedness of the polynomial pn(x) for any n. Indeed, assume that ξ is a non-real root of pn(x), and let x = ξ and arXiv:1409.2527v1 [math.CO] 8 Sep 2014 y = ξ.
    [Show full text]
  • The Matching Polynomial of a Distance-Regular Graph
    Internat. J. Math. & Math. Sci. Vol. 23, No. 2 (2000) 89–97 S0161171200000740 ©Hindawi Publishing Corp. THE MATCHING POLYNOMIAL OF A DISTANCE-REGULAR GRAPH ROBERT A. BEEZER and E. J. FARRELL (Received 15 August 1997) Abstract. A distance-regular graph of diameter d has 2d intersection numbers that de- termine many properties of graph (e.g., its spectrum). We show that the first six coefficients of the matching polynomial of a distance-regular graph can also be determined from its intersection array, and that this is the maximum number of coefficients so determined. Also, the converse is true for distance-regular graphs of small diameter—that is, the inter- section array of a distance-regular graph of diameter 3 or less can be determined from the matching polynomial of the graph. Keywords and phrases. Matching polynomial, distance-regular graph. 2000 Mathematics Subject Classification. Primary 05C70, 05E30. 1. Introduction. Distance-regular graphs are highly regular combinatorial struc- tures that often occur in connection with other areas of combinatorics (e.g., designs and finite geometries) and many of their properties can be determined from their intersection numbers. These properties include the eigenvalues of the graph, and their multiplicities, and hence, we can determine the characteristic polynomial of a distance-regular graph from knowledge of its intersection numbers. It is also known that the characteristic polynomial of any graph can be determined by computing the circuit polynomial and converting it to a polynomial in a single variable via a specific set of substitutions. Since the intersection array and the circuit polynomial of a distance-regular graph both determine its characteristic polynomial, it was natural to investigate the relation- ship between the circuit polynomial and the intersection array for distance-regular graphs.
    [Show full text]
  • Combinatorics and Zeros of Multivariate Polynomials
    Combinatorics and zeros of multivariate polynomials NIMA AMINI Doctoral Thesis in Mathematics Stockholm, Sweden 2019 TRITA-SCI-FOU 2019:33 KTH School of Engineering Sciences ISRN KTH/MAT/A-19/05-SE SE-100 44 Stockholm ISBN 978-91-7873-210-4 SWEDEN Akademisk avhandling som med tillst˚andav Kungl Tekniska h¨ogskolan framl¨agges till offentlig granskning f¨or avl¨aggande av filosofie doktorsexamen i matematik fredagen den 24 maj 2019 klockan 14.00 i sal D3, Lindstedtsv¨agen 5, KTH, Stock- holm. c Nima Amini, 2019 Tryck: Universitetsservice US-AB, 2019 iii Abstract This thesis consists of five papers in algebraic and enumerative combina- torics. The objects at the heart of the thesis are combinatorial polynomials in one or more variables. We study their zeros, coefficients and special eval- uations. Hyperbolic polynomials may be viewed as multivariate generalizations of real-rooted polynomials in one variable. To each hyperbolic polynomial one may associate a convex cone from which a matroid can be derived - a so called hyperbolic matroid. In Paper A we prove the existence of an infinite family of non-representable hyperbolic matroids parametrized by hypergraphs. We further use special members of our family to investigate consequences to a cen- tral conjecture around hyperbolic polynomials, namely the generalized Lax conjecture. Along the way we strengthen and generalize several symmetric function inequalities in the literature, such as the Laguerre-Tur´aninequality and an inequality due to Jensen. In Paper B we affirm the generalized Lax conjecture for two related classes of combinatorial polynomials: multivariate matching polynomials over arbitrary graphs and multivariate independence polynomials over simplicial graphs.
    [Show full text]
  • Lecture 16: Roots of the Matching Polynomial 16.1 Real Rootedness
    Counting and Sampling Fall 2017 Lecture 16: Roots of the Matching Polynomial Lecturer: Shayan Oveis Gharan November 22nd Disclaimer: These notes have not been subjected to the usual scrutiny reserved for formal publications. Recall that for a graph G n=2 X k n−2k µG(x) = (−1) mkx : k=0 In this sectionp we prove that the matching polynomial is real rooted and all of its roots are bounded from above by 2 ∆ − 1 assuming that the maximum degree of G is ∆. 16.1 Real Rootedness of Matching Polynomial + The following fact can be proven similar to the facts on µG(x) that we proved in lecture 12. Fact 16.1. For any pair of disjoint graphs G; H µG[H (x) = µG(x) · µH (x): For any graph G and any vertex u, X µG(x) = xµG−u(x) − µG−u−v(x): v∼u For a graph G and a vertex u, the path-tree of G with respect to u, T = TG(u) is defined as follows: For every path in G that starts at u, T has a node and two paths are adjacent if their length differs by 1 and one is a prefix of another. See the following figure for an example: 012 0123 01234 01 0134 1 013 0132 0 0 2 3 4 021 0213 02134 02 0231 023 0234 We prove the following theorem due to Godsil and Gutman. Theorem 16.2. Let G be a graph and u be a vertex of G. Also, let T = T (G; u) be the path tree of G with respect to u.
    [Show full text]
  • Polynomial Reconstruction of the Matching Polynomial
    Electronic Journal of Graph Theory and Applications 3 (1) (2015), 27–34 Polynomial reconstruction of the matching polynomial Xueliang Li, Yongtang Shi, Martin Trinks Center for Combinatorics, Nankai University, Tianjin 300071, China [email protected], [email protected], [email protected] Abstract The matching polynomial of a graph is the generating function of the numbers of its matchings with respect to their cardinality. A graph polynomial is polynomial reconstructible, if its value for a graph can be determined from its values for the vertex-deleted subgraphs of the same graph. This note discusses the polynomial reconstructibility of the matching polynomial. We collect previous results, prove it for graphs with pendant edges and disprove it for some graphs. Keywords: reconstruction, matching polynomial, perfect matching Mathematics Subject Classification : 05C31, 05C70 DOI: 10.5614/ejgta.2015.3.1.4 1. Introduction The famous (and still unsolved) reconstruction conjecture of Kelly [9] and Ulam [15] states that every graph G with at least three vertices can be reconstructed from (the isomorphism classes of) its vertex-deleted subgraphs. With respect to a graph polynomial P (G), this question may be adapted as follows: Can P (G) of a graph G = (V; E) be reconstructed from the graph polynomials of the vertex deleted- subgraphs, that is from the collection P (G−v) for v V ? Here, this problem is considered for the 2 matching polynomial of a graph, which is the generating function of the number of its matchings with respect to their cardinality. This paper aims to prove that graphs with pendant edges are polynomial reconstructible and, on the other hand, to display some evidence that arbitrary graphs are not.
    [Show full text]
  • On Weakly Distinguishing Graph Polynomials 3
    Discrete Mathematics and Theoretical Computer Science DMTCS vol. 21:1, 2019, #4 On Weakly Distinguishing Graph Polynomials Johann A. Makowsky1 Vsevolod Rakita2 1 Technion - Israel Institute of Technology, Department of Computer Science 2 Technion - Israel Institute of Technology, Department of Mathematics received 1st Nov. 2018, revised 5th Mar. 2019, accepted 25th Mar. 2019. A univariate graph polynomial P (G; X) is weakly distinguishing if for almost all finite graphs G there is a finite graph H with P (G; X) = P (H; X). We show that the clique polynomial and the independence polynomial are weakly distinguishing. Furthermore, we show that generating functions of induced subgraphs with property C are weakly distinguishing provided that C is of bounded degeneracy or treewidth. The same holds for the harmonious chromatic polynomial. Keywords: Graph Theory, Graph Polynomials 1 Introduction and Outline Throughout this paper we consider only simple (i.e. finite, undirected loopless graphs without parallel edges), vertex labelled graphs. Let P be a graph polynomial. A graph G is P -unique if every graph H with P (G; X) = P (H; X) is isomorphic to G. A graph H is a P -mate of G if P (G; X) = P (H; X) but H is not isomorphic to G. In [14] P -unique graphs are studied for the Tutte polynomial T (G; X, Y ), the chromatic polynomial χ(G; X), the matching polynomial m(G; X) and the characteristic polynomial char(P ; X). A statement holds for almost all graphs if the proportion of graphs of order n for which it holds, tends to 1, when n tends to infinity.
    [Show full text]