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 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 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.

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.

Greedy Algorithms

1995 ◽  
Author(s):  
Raymond Greenlaw ◽  
H. James Hoover ◽  
Walter L. Ruzzo
Keyword(s):  
Greedy Algorithms ◽  
Independent Set ◽  
Independent Sets ◽  
Case Finding ◽  
Sequential Algorithm ◽  
Maximal Independent Set ◽  
Maximum Independent Set ◽  
Greedy Method ◽  
Numerical Order ◽  
Selection Of

We consider the selection of two basketball teams at a neighborhood playground to illustrate the greedy method. Usually the top two players are designated captains. All other players line up while the captains alternate choosing one player at a time. Usually, the players are picked using a greedy strategy. That is, the captains choose the best unclaimed player. The system of selection of choosing the best, most obvious, or most convenient remaining candidate is called the greedy method. Greedy algorithms often lead to easily implemented efficient sequential solutions to problems. Unfortunately, it also seems to be that sequential greedy algorithms frequently lead to solutions that are inherently sequential — the solutions produced by these algorithms cannot be duplicated rapidly in parallel, unless NC equals P. In the following subsections we will examine this phenomenon. We illustrate some of the important aspects of greedy algorithms using one that constructs a maximal independent set in a graph. An independent set is a set of vertices of a graph that are pairwise nonadjacent. A maximum independent set is such a set of largest cardinality. It is well known that finding maximum independent sets is NP-hard. An independent set is maximal if no other vertex can be added while maintaining the independent set property. In contrast to the maximum case, finding maxima? independent sets is very easy. Figure 7.1.1 depicts a simple polynomial time sequential algorithm computing a maximal independent set. The algorithm is a greedy algorithm: it processes the vertices in numerical order, always attempting to add the lowest numbered vertex that has not yet been tried. The sequential algorithm in Figure 7.1.1, having processed vertices 1,... , j -1, can easily decide whether to include vertex j. However, notice that its decision about j potentially depends on its decisions about all earlier vertices — j will be included in the maximal independent set if and only if all j' less than j and adjacent to it were excluded.

scholarly journals Max-independent set and the quantum alternating operator ansatz

2020 ◽  
Vol 18 (04) ◽  
pp. 2050011 ◽  
Author(s):  
Zain Hamid Saleem
Keyword(s):  
Graph Theory ◽  
Probability Distribution ◽  
Independent Set ◽  
Independent Sets ◽  
Maximum Independent Set ◽  
Initial State ◽  
Max Cut Problem ◽  
Initial States ◽  
Circuit Depth ◽  
Ring Graph

The maximum-independent set (MIS) problem of graph theory using the quantum alternating operator ansatz is studied. We perform simulations on the Rigetti Forest simulator for the square ring, [Formula: see text], and [Formula: see text] graphs and analyze the dependence of the algorithm on the depth of the circuit and initial states. The probability distribution of observation of the feasible states representing maximum-independent sets is observed to be asymmetric for the MIS problem, which is unlike the Max-Cut problem where the probability distribution of feasible states is symmetric. For asymmetric graphs, it is shown that the algorithm clearly favors the independent set with the larger number of elements even for finite circuit depth. We also compare the approximation ratios for the algorithm when we choose different initial states for the square ring graph and show that it is dependent on the choice of the initial state.

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 Gumbel-softmax-based optimization: a simple general framework for optimization problems on graphs

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Yaoxin Li ◽  
Jing Liu ◽  
Guozheng Lin ◽  
Yueyuan Hou ◽  
Muyun Mou ◽  
...  
Keyword(s):  
Objective Function ◽  
Statistical Physics ◽  
Automatic Differentiation ◽  
Optimization Problems ◽  
Vertex Cover ◽  
Independent Set ◽  
Computational Social Science ◽  
Maximization Problem ◽  
Maximum Independent Set ◽  
Gradient Descent Algorithm

AbstractIn computer science, there exist a large number of optimization problems defined on graphs, that is to find a best node state configuration or a network structure, such that the designed objective function is optimized under some constraints. However, these problems are notorious for their hardness to solve, because most of them are NP-hard or NP-complete. Although traditional general methods such as simulated annealing (SA), genetic algorithms (GA), and so forth have been devised to these hard problems, their accuracy and time consumption are not satisfying in practice. In this work, we proposed a simple, fast, and general algorithm framework based on advanced automatic differentiation technique empowered by deep learning frameworks. By introducing Gumbel-softmax technique, we can optimize the objective function directly by gradient descent algorithm regardless of the discrete nature of variables. We also introduce evolution strategy to parallel version of our algorithm. We test our algorithm on four representative optimization problems on graph including modularity optimization from network science, Sherrington–Kirkpatrick (SK) model from statistical physics, maximum independent set (MIS) and minimum vertex cover (MVC) problem from combinatorial optimization on graph, and Influence Maximization problem from computational social science. High-quality solutions can be obtained with much less time-consuming compared to the traditional approaches.

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