scholarly journals Generalizations of the Matching Polynomial to the Multivariate Independence Polynomial

2019 ◽  
Vol 2 (5) ◽  
pp. 781-802
Author(s):  
Jonathan D. Leake ◽  
Nick R. Ryder
Keyword(s):  
Matching Polynomial ◽  
Independence Polynomial

Independence polynomial and matching polynomial of the Koch network

2015 ◽  
Vol 29 (32) ◽  
pp. 1550234
Author(s):  
Yunhua Liao ◽  
Xiaoliang Xie
Keyword(s):  
Lattice Gas ◽  
Small World ◽  
Partition Functions ◽  
Complete Class ◽  
Dimer Model ◽  
Scale Free ◽  
Matching Polynomial ◽  
Independence Polynomial ◽  
Classical Models ◽  
Explicit Formulae

The lattice gas model and the monomer-dimer model are two classical models in statistical mechanics. It is well known that the partition functions of these two models are associated with the independence polynomial and the matching polynomial in graph theory, respectively. Both polynomials have been shown to belong to the “[Formula: see text]-complete” class, which indicate the problems are computationally “intractable”. We consider these two polynomials of the Koch networks which are scale-free with small-world effects. Explicit recurrences are derived, and explicit formulae are presented for the number of independent sets of a certain type.

scholarly journals A new two-variable generalization of the chromatic polynomial

2003 ◽  
Vol Vol. 6 no. 1 ◽  
Author(s):  
Klaus Dohmen ◽  
André Poenitz ◽  
Peter Tittmann
Keyword(s):  
Chromatic Polynomial ◽  
Complete Graphs ◽  
Matching Polynomial ◽  
Independence Polynomial ◽  
Decomposition Formula ◽  
International Audience ◽  
Complete Bipartite Graphs ◽  
Chromatic Symmetric Function ◽  
Broken Circuit ◽  
Edge Decomposition

International audience We present a two-variable polynomial, which simultaneously generalizes the chromatic polynomial, the independence polynomial, and the matching polynomial of a graph. This new polynomial satisfies both an edge decomposition formula and a vertex decomposition formula. We establish two general expressions for this new polynomial: one in terms of the broken circuit complex and one in terms of the lattice of forbidden colorings. We show that the new polynomial may be considered as a specialization of Stanley's chromatic symmetric function. We finally give explicit expressions for the generalized chromatic polynomial of complete graphs, complete bipartite graphs, paths, and cycles, and show that it can be computed in polynomial time for trees and graphs of restricted pathwidth.

scholarly journals On the Independence Polynomial of an Antiregular Graph

2012 ◽  
Vol 28 (2) ◽  
pp. 279-288
Author(s):  
VADIM E. LEVIT ◽  
◽  
EUGEN MANDRESCU ◽  
Keyword(s):  
Graph Theory ◽  
Independent Set ◽  
Independent Sets ◽  
Maximum Independent Set ◽  
Real Roots ◽  
Matching Polynomial ◽  
Independence Polynomial ◽  
Independence Polynomials ◽  
Threshold Graphs ◽  
The Family

A graph with at most two vertices of the same degree is known as antiregular [ Merris, R., Antiregular graphs are universal for trees, Publ. Electrotehn. Fak. Univ. Beograd, Ser. Mat. 14 (2003) 1-3], maximally nonregular [Zykov, A. A., Fundamentals of graph theory, BCS Associates, Moscow, 1990] or quasiperfect [ Behzad, M. and Chartrand, D. M., No graph is perfect, Amer. Math. Monthly 74 (1967), 962-963]. If sk is the number of independent sets of cardinality k in a graph G, then I(G; x) = s0 +s1x+...+sαx α is the independence polynomial of G [ Gutman, I. and Harary, F., Generalizations of the matching polynomial, Utilitas Mathematica 24 (1983), 97-106] , where α = α(G) is the size of a maximum independent set. In this paper we derive closed formulas for the independence polynomials of antiregular graphs. It results in proving that every antiregular graph is uniquely defined by its independence polynomial within the family of threshold graphs. Moreover, the independence polynomial of each antiregular graph is log-concave, it has two real roots at most, and its value at −1 belongs to {−1, 0}.

scholarly journals INDEPENDENCE AND PI POLYNOMIALS FOR FEW STRINGS

2019 ◽  
pp. 761
Author(s):  
Amir Loghman ◽  
Mahtab Khanlar Motlagh
Keyword(s):  
Molecular Graph ◽  
Independent Set ◽  
Independent Sets ◽  
Topological Indices ◽  
Maximum Independent Set ◽  
Matching Polynomial ◽  
Independence Polynomial ◽  
Independence Polynomials

If $s_k$ is the number of independent sets of cardinality $k$ in a graph $G$, then $I(G; x)= s_0+s_1x+…+s_{\alpha} x^{\alpha}$ is the independence polynomial of $G$ [ Gutman, I. and Harary, F., Generalizations of the matching polynomial, Utilitas Mathematica 24 (1983) 97-106] , where $\alpha=\alpha(G)$ is the size of a maximum independent set. Also the PI polynomial of a molecular graph $G$ is defined as $A+\sum x^{|E(G)|-N(e)}$, where $N(e)$ is the number of edges parallel to $e$, $A=|V(G)|(|V(G)|+1)/2-|E(G)|$ and summation goes over all edges of $G$. In [T. Do$\check{s}$li$\acute{c}$, A. Loghman and L. Badakhshian, Computing Topological Indices by Pulling a Few Strings, MATCH Commun. Math. Comput. Chem. 67 (2012) 173-190], several topological indices for all graphs consisting of at most three strings are computed. In this paper we compute the PI and independence polynomials for graphs containing one, two and three strings.

scholarly journals The Complexity of Approximating the Matching Polynomial in the Complex Plane

2021 ◽  
Vol 13 (2) ◽  
pp. 1-37
Author(s):  
Ivona Bezáková ◽  
Andreas Galanis ◽  
Leslie Ann Goldberg ◽  
Daniel Štefankovič
Keyword(s):  
Real Axis ◽  
Complex Plane ◽  
Maximum Degree ◽  
Negative Real Axis ◽  
Positive Real ◽  
Bounded Degree ◽  
Matching Polynomial ◽  
Correlation Decay ◽  
Connective Constant ◽  
General Complex

We study the problem of approximating the value of the matching polynomial on graphs with edge parameter γ, where γ takes arbitrary values in the complex plane. When γ is a positive real, Jerrum and Sinclair showed that the problem admits an FPRAS on general graphs. For general complex values of γ, Patel and Regts, building on methods developed by Barvinok, showed that the problem admits an FPTAS on graphs of maximum degree Δ as long as γ is not a negative real number less than or equal to −1/(4(Δ −1)). Our first main result completes the picture for the approximability of the matching polynomial on bounded degree graphs. We show that for all Δ ≥ 3 and all real γ less than −1/(4(Δ −1)), the problem of approximating the value of the matching polynomial on graphs of maximum degree Δ with edge parameter γ is #P-hard. We then explore whether the maximum degree parameter can be replaced by the connective constant. Sinclair et al. showed that for positive real γ, it is possible to approximate the value of the matching polynomial using a correlation decay algorithm on graphs with bounded connective constant (and potentially unbounded maximum degree). We first show that this result does not extend in general in the complex plane; in particular, the problem is #P-hard on graphs with bounded connective constant for a dense set of γ values on the negative real axis. Nevertheless, we show that the result does extend for any complex value γ that does not lie on the negative real axis. Our analysis accounts for complex values of γ using geodesic distances in the complex plane in the metric defined by an appropriate density function.

scholarly journals On the independence polynomial of the corona of graphs

2016 ◽  
Vol 203 ◽  
pp. 85-93 ◽  
Author(s):  
Vadim E. Levit ◽  
Eugen Mandrescu
Keyword(s):  
Independence Polynomial

MATCHING POLYNOMIALS OF SERIES-PARALLEL GRAPHS

1993 ◽  
Vol 03 (01) ◽  
pp. 13-18 ◽  
Author(s):  
LIH-HSING HSU
Keyword(s):  
Parallel Algorithm ◽  
Efficient Algorithm ◽  
Matching Polynomial ◽  
Erew Pram ◽  
Parallel Graph

We present an efficient algorithm for computing the matching polynomial of a series-parallel graph in O(n2) time. This algorithm improves on the previous result of O(n3). We also present a cost-optimal parallel algorithm for computing the matching polynomial of a series-parallel graph using an EREW PRAM computer with the number of processors p less than n2/ log n.

scholarly journals The independence polynomial of rooted products of graphs

2010 ◽  
Vol 158 (5) ◽  
pp. 551-558 ◽  
Author(s):  
Vladimir R. Rosenfeld
Keyword(s):  
Independence Polynomial

Combinatorial aspects of the Löwenstein avoidance rule. Part I: the independence polynomial

2021 ◽  
Vol 77 (6) ◽  
Author(s):  
Montauban Moreira de Oliveira Jr ◽  
Jean-Guillaume Eon
Keyword(s):  
Propositional Calculus ◽  
Independent Sets ◽  
Companion Paper ◽  
Theoretical Interpretation ◽  
Vector Method ◽  
Independence Polynomial ◽  
Calculation Technique ◽  
Independence Polynomials ◽  
Maximal Independent Sets

According to Löwenstein's rule, Al–O–Al bridges are forbidden in the aluminosilicate framework of zeolites. A graph-theoretical interpretation of the rule, based on the concept of independent sets, was proposed earlier. It was shown that one can apply the vector method to the associated periodic net and define a maximal Al/(Al+Si) ratio for any aluminosilicate framework following the rule; this ratio was called the independence ratio of the net. According to this method, the determination of the independence ratio of a periodic net requires finding a subgroup of the translation group of the net for which the quotient graph and a fundamental transversal have the same independence ratio. This article and a companion paper deal with practical issues regarding the calculation of the independence ratio of mainly 2-periodic nets and the determination of site distributions realizing this ratio. The first paper describes a calculation technique based on propositional calculus and introduces a multivariate polynomial, called the independence polynomial. This polynomial can be calculated in an automatic way and provides the list of all maximal independent sets of the graph, hence also the value of its independence ratio. Some properties of this polynomial are discussed; the independence polynomials of some simple graphs, such as short paths or cycles, are determined as examples of calculation techniques. The method is also applied to the determination of the independence ratio of the 2-periodic net dhc.

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