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.

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.

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.

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 The independence and clique polynomial of the conjugacy class graph of dihedral group

2018 ◽  
Vol 14 ◽  
pp. 434-438
Author(s):  
Nabilah Najmuddin ◽  
Nor Haniza Sarmin ◽  
Ahmad Erfanian ◽  
Hamisan Rahmat
Keyword(s):  
Conjugacy Class ◽  
Complete Graph ◽  
Dihedral Group ◽  
Independent Set ◽  
Independent Sets ◽  
Conjugacy Classes ◽  
Dihedral Groups ◽  
Independence Polynomial ◽  
Independence Polynomials ◽  
Class Graph

The independence and clique polynomial are two types of graph polynomial that store combinatorial information of a graph. The independence polynomial of a graph is the polynomial in which its coefficients are the number of independent sets in the graph. The independent set of a graph is a set of vertices that are not adjacent. The clique polynomial of a graph is the polynomial in which its coefficients are the number of cliques in the graph. The clique of a graph is a set of vertices that are adjacent. Meanwhile, a graph of group G is called conjugacy class graph if the vertices are non-central conjugacy classes of G and two distinct vertices are connected if and only if their class cardinalities are not coprime. The independence and clique polynomial of the conjugacy class graph of a group G can be obtained by considering the polynomials of complete graph or polynomials of union of some graphs. In this research, the independence and clique polynomials of the conjugacy class graph of dihedral groups of order 2n are determined based on three cases namely when n is odd, when n and n/2 are even, and when n is even and n/2 is odd. For each case, the results of the independence polynomials are of degree two, one and two, and the results of the clique polynomials are of degree (n-1)/2, (n+2)/2 and (n-2)/2, respectively.

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 Christoffel–Darboux Type Identities for the Independence Polynomial

2018 ◽  
Vol 27 (5) ◽  
pp. 716-724
Author(s):  
FERENC BENCS
Keyword(s):  
Free Graph ◽  
Real Roots ◽  
Independence Polynomial ◽  
Independence Polynomials

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 Chudnovsky and Seymour, which states that the independence polynomial of a claw-free graph has only real roots. Another application is related to a conjecture of Merrifield and Simmons.

scholarly journals Polynomials with Real Zeros and Compatible Sequences

10.37236/2674 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Li Liu
Keyword(s):  
Simple Proof ◽  
Opposite Sign ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Real Zeros ◽  
Independence Polynomial ◽  
Necessary And Sufficient ◽  
Interlacing Property

In this paper, we study polynomials with only real zeros based on the method of compatible zeros. We obtain a necessary and sufficient condition for the compatible property of two polynomials whose leading coefficients have opposite sign. As applications, we partially answer a question proposed by M. Chudnovsky and P. Seymour in the recent publication [M. Chudnovsky, P. Seymour, The roots of the independence polynomial of a clawfree graph, J. Combin. Theory Ser. B 97 (2007) 350--357]. We also establish the connection between the interlacing property and the compatible property of two polynomials and give a simple proof of some known results.

scholarly journals On the Stability of Independence Polynomials

10.37236/7280 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Jason I. Brown ◽  
Ben Cameron
Keyword(s):  
Half Plane ◽  
Complex Analysis ◽  
Independence Number ◽  
Independent Sets ◽  
Induced Subgraph ◽  
Left Half ◽  
Independence Polynomial ◽  
Independence Polynomials ◽  
The Stability ◽  
The Right

The independence polynomial of a graph is the generating polynomial for the number of independent sets of each size and its roots are called independence roots. We investigate the stability of such polynomials, that is, conditions under which the independence roots lie in the left half-plane. We use results from complex analysis to determine graph operations that result in a stable or nonstable independence polynomial. In particular, we prove that every graph is an induced subgraph of a graph with stable independence polynomial. We also show that the independence polynomials of graphs with independence number at most three are necessarily stable, but for larger independence number, we show that the independence polynomials can have roots arbitrarily far to the right.

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.

Sign in / Sign up

Export Citation Format

Share Document