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.

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.

The Number of Zeros of a Polynomial in a Half-Plane

1960 ◽  
Vol 56 (2) ◽  
pp. 132-147 ◽  
Author(s):  
A. Talbot
Keyword(s):  
Half Plane ◽  
Continued Fraction ◽  
Complex Polynomial ◽  
Continued Fraction Expansion ◽  
Left Half ◽  
Number Of Zeros ◽  
Full Discussion ◽  
The Subject ◽  
The Right

The determination of the number of zeros of a complex polynomial in a half-plane, in particular in the upper and lower, or right and left, half-planes, has been the subject of numerous papers, and a full discussion, with many references, is given in Marden (l) and Wall (2), where the basis for the determination is a continued-fraction expansion, or H.C.F. algorithm, in terms of which the number of zeros in one of the half-planes can be written down at once. In addition, determinantal formulae for the relevant elements of the algorithm can be obtained, and these lead to determinantal criteria for the number of zeros, including that of Hurwitz (3) for the right and left half-planes.

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 Singular Values of Two Parameter Families $\lambda\dfrac{e^{az}-1}{z}$ and $\lambda\dfrac{z}{e^{az}-1}$

2016 ◽  
Vol 8 (1) ◽  
pp. 10 ◽  
Author(s):  
Mohammad Sajid
Keyword(s):  
Half Plane ◽  
Meromorphic Functions ◽  
Entire Functions ◽  
Singular Values ◽  
Critical Values ◽  
Open Disk ◽  
Left Half ◽  
Left Half Plane ◽  
The Right ◽  
Two Parameter

<p>The singular values of two parameter families of entire functions $f_{\lambda,a}(z)=\lambda\frac{e^{az}-1}{z}$, $f_{\lambda,a}(0)=\lambda a$ and meromorphic functions $g_{\lambda,a}(z)=\lambda\frac{z}{e^{az}-1}$, $g_{\lambda,a}(0)=\frac{\lambda}{a}$, $\lambda, a \in \mathbb{R} \backslash \{0\}$, $z \in \mathbb{C}$, are investigated. It is shown that all the critical values of $f_{\lambda,a}(z)$ and $g_{\lambda,a}(z)$ lie in the right half plane for $a&lt;0$ and lie in the left half plane for $a&gt;0$. It is described that the functions $f_{\lambda,a}(z)$ and $g_{\lambda,a}(z)$ have infinitely many singular values. It is also found that all the singular values $f_{\lambda,a}(z)$ are bounded and lie inside the open disk centered at origin and having radius $|\lambda a|$ and all the critical values of $g_{\lambda,a}(z)$ belong to the exterior of the disk centered at origin and having radius $|\frac{\lambda}{a}|$.</p>

scholarly journals Location of eigenmodes of Euler–Bernoulli beam model under fully non-dissipative boundary conditions

2019 ◽  
Vol 475 (2231) ◽  
pp. 20190544
Author(s):  
Marianna A. Shubov
Keyword(s):  
Boundary Conditions ◽  
Half Plane ◽  
Adjoint Problem ◽  
Beam Model ◽  
Control Parameters ◽  
Bernoulli Beam ◽  
Left Half ◽  
The Right ◽  
Euler Bernoulli Beam ◽  
Bernoulli Beam Model

The Euler–Bernoulli beam model with non-conservative feedback-type boundary conditions is investigated. Components of the two-dimensional input vector are shear and moment at the right end, and components of the observation vector are time derivative of displacement and slope at the right end. The boundary matrix containing four control parameters relates input and observation. The following results are presented: (i) if one and only one of the control parameters is positive and the rest of them are equal to zero, then the set of the eigenmodes is located in the open left half-plane of the complex plane, which means that all eigenmodes are stable; (ii) if the diagonal elements of the boundary matrix are positive and off-diagonal elements are zeros, then the set of the eigenmodes is located in the open left half-plane, which implies stability of all eigenmodes; (iii) specific combinations of the diagonal and off-diagonal elements have been found to ensure the stability results. To prove the results, two special relations between the eigenmodes and mode shapes of the non-self-adjoint problem and clamped–free self-adjoint problem have been established.

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.

Symmetrizable Difference Dchemes

2005 ◽  
Vol 5 (1) ◽  
pp. 3-50 ◽  
Author(s):  
Alexei A. Gulin
Keyword(s):  
Initial Data ◽  
Difference Scheme ◽  
Difference Schemes ◽  
Time Dependent ◽  
Stability Boundaries ◽  
Typical Representative ◽  
Variable Weight ◽  
The Stability ◽  
Conditions Of Stability ◽  
The Right

AbstractA review of the stability theory of symmetrizable time-dependent difference schemes is represented. The notion of the operator-difference scheme is introduced and general ideas about stability in the sense of the initial data and in the sense of the right hand side are formulated. Further, the so-called symmetrizable difference schemes are considered in detail for which we manage to formulate the unimprovable necessary and su±cient conditions of stability in the sense of the initial data. The schemes with variable weight multipliers are a typical representative of symmetrizable difference schemes. For such schemes a numerical algorithm is proposed and realized for constructing stability boundaries.

Pemanfaatan Metil Ester Sulfonat pada Pembuatan Deterjen Cair

2016 ◽  
Vol 7 (14) ◽  
pp. 143-155
Author(s):  
Eldha Sampepana ◽  
Suroto Hadi Saputra
Keyword(s):  
Fatty Acids ◽  
Methyl Ester ◽  
Linear Alkylbenzene Sulfonate ◽  
Emulsion Stability ◽  
Alcohol Sulfate ◽  
Stable Emulsion ◽  
Linear Alkylbenzene ◽  
Alkylbenzene Sulfonate ◽  
The Stability ◽  
The Right

In the manufacture of detergents still using surfactants (which serves as an emulsifier) of crude oil in the form of the AS. (alcohol sulfate) and LAS (linear alkylbenzene sulfonate), where this type of surfactant cannot be degraded by microorganisms when discharged into the environment, causing environmental pollution. Methyl ester sulfonate surfactant is an anionic surfactant which has a composition of C16 - C18 fatty acids are capable of acting against nature deterjensinya, while the C12 - C14 fatty acids contribute to the foaming effect. The purpose of this study was to look for the formulation of methyl ester sulfonate (MES) the right to produce a good detergent by using materials such as methyl ester sulfonate surfactant self-made, methyl ester sulfonate and sodium lauryl market Ester Sulfate (SLS) with a concentration of 15 %, 20 % and 25 %. Detergent results of the study have high detergency ( net ) compared with the detergency of detergent commercial, have a stable emulsion stability, the stability of the foam/foam detergent power made from methyl ester sulfonate surfactant produces less foam, compared with a detergent made from SLS and surfactant SNI 06-4075-1996 standards.

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