scholarly journals Independence polynomial of the commuting and noncommuting graphs associated to the quasidihedral group

2021 ◽  
Vol 1988 (1) ◽  
pp. 012072
Author(s):  
Nabilah Najmuddin ◽  
Nor Haniza Sarmin ◽  
Ahmad Erfanian
Keyword(s):  
Independence Polynomial

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

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 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.

Unimodality of independence polynomials of rooted products of graphs

2019 ◽  
Vol 150 (5) ◽  
pp. 2573-2585
Author(s):  
Bao-Xuan Zhu ◽  
Qingxiu Wang
Keyword(s):  
Real Zeros ◽  
Independence Polynomial ◽  
Independence Polynomials

AbstractIn 1987, Alavi, Malde, Schwenk and Erdős conjectured that the independence polynomial of any tree is unimodal. Although it attracts many researchers' attention, it is still open. Motivated by this conjecture, in this paper, we prove that rooted products of some graphs preserve real rootedness of independence polynomials. As application, we not only give a unified proof for some known results, but also we can apply them to generate infinite kinds of trees whose independence polynomials have only real zeros. Thus their independence polynomials are unimodal.

On the coefficients of the independence polynomial of graphs

2016 ◽  
Vol 33 (4) ◽  
pp. 1324-1342 ◽  
Author(s):  
Shuchao Li ◽  
Lin Liu ◽  
Yueyu Wu
Keyword(s):  
Independence Polynomial

scholarly journals The independence polynomial of inverse commuting graph of dihedral groups

2020 ◽  
Vol 16 (1) ◽  
pp. 115-120
Author(s):  
Aliyu Suleiman ◽  
Aliyu Ibrahim Kiri
Keyword(s):  
Identity Element ◽  
Independent Set ◽  
Independent Sets ◽  
Dihedral Groups ◽  
Independence Polynomial ◽  
Independence Polynomials ◽  
Commuting Graph ◽  
Central Elements ◽  
Vertex Set

Set of vertices not joined by an edge in a graph is called the independent set of the graph. The independence polynomial of a graph is a polynomial whose coefficient is the number of independent sets in the graph. In this research, we introduce and investigate the inverse commuting graph of dihedral groups (D2N) denoted by GIC. It is a graph whose vertex set consists of the non-central elements of the group and for distinct  x,y, E D2N, x and y are adjacent if and only if xy = yx = 1  where 1 is the identity element. The independence polynomials of the inverse commuting graph for dihedral groups are also computed. A formula for obtaining such polynomials without getting the independent sets is also found, which was used to compute for dihedral groups of order 18 up to 32.

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 Unimodality of the independence polynomials of some composite graphs

Filomat ◽  
2017 ◽  
Vol 31 (3) ◽  
pp. 629-637 ◽  
Author(s):  
Bao-Xuan Zhu ◽  
Qinglin Lu
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Lexicographic Product ◽  
Complete Multipartite Graph ◽  
Real Zeros ◽  
Independence Polynomial ◽  
Independence Polynomials ◽  
Multipartite Graph ◽  
Complete Multipartite

Let I(G;x) denote the independence polynomial of a graph G. In this paper we study the unimodality properties of I(G; x) for some composite graphs G. Given two graphs G1 and G2, let G1[G2] denote the lexicographic product of G1 and G2. Assume I(G1; x) = P i_0 aixi and I(G2; x) = P i_0 bixi, where I(G2; x) is log-concave. Then we prove (i) if I(G1; x) is logconcave and (a2i ??ai??1ai+1)b21 _ aiai??1b2 for all 1 _ i _ _(G1), then I(G1[G2]; x) is log-concave; (ii) if ai??1 _ b1ai for 1 _ i _ _(G1), then I(G1[G2]; x) is unimodal. In particular, if ai is increasing in i, then I(G1[G2]; x) is unimodal. We also give two su_cient conditions when the independence polynomial of a complete multipartite graph is unimodal or log-concave. Finally, for every odd positive integer _ > 3, we find a connected graph G not a tree, such that _(G) = _, and I(G; x) is symmetric and has only real zeros. This answers a problem of Mandrescu and Miric?a.

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