Matching polynomials for some nanostar dendrimers

2021 ◽  
pp. 2150188
Author(s):  
Fateme movahedi
Keyword(s):  
Hosoya Index ◽  
Matching Polynomial ◽  
Recursive Formulas

Dendrimers are highly branched monodisperse, macromolecules and are considered in nanotechnology with a variety of suitable applications. In this paper, the matching polynomial and some results of the matchings for three classes of nanostar dendrimers are obtained. Furthermore, we express the recursive formulas of the Hosoya index for these structures of dendrimers by their matching polynomials.

scholarly journals The higher-order matching polynomial of a graph

2005 ◽  
Vol 2005 (10) ◽  
pp. 1565-1576 ◽  
Author(s):  
Oswaldo Araujo ◽  
Mario Estrada ◽  
Daniel A. Morales ◽  
Juan Rada
Keyword(s):  
Hypergeometric Functions ◽  
Higher Order ◽  
Disjoint Paths ◽  
Hosoya Index ◽  
Matching Polynomial ◽  
Similarities And Differences

Given a graphGwithnvertices, letp(G,j)denote the number of waysjmutually nonincident edges can be selected inG. The polynomialM(x)=∑j=0[n/2](−1)jp(G,j)xn−2j, called the matching polynomial ofG, is closely related to the Hosoya index introduced in applications in physics and chemistry. In this work we generalize this polynomial by introducing the number of disjoint paths of lengtht, denoted bypt(G,j). We compare this higher-order matching polynomial with the usual one, establishing similarities and differences. Some interesting examples are given. Finally, connections between our generalized matching polynomial and hypergeometric functions are found.

scholarly journals Two Classes of Topological Indices of Phenylene Molecule Graphs

2016 ◽  
Vol 2016 ◽  
pp. 1-5 ◽  
Author(s):  
Tingzeng Wu
Keyword(s):  
Topological Indices ◽  
Extremal Graph ◽  
Hosoya Index ◽  
Matching Polynomial ◽  
Conjugated Hydrocarbons ◽  
Independent Edge ◽  
The Absolute

A phenylene is a conjugated hydrocarbons molecule composed of six- and four-membered rings. The matching energy of a graphGis equal to the sum of the absolute values of the zeros of the matching polynomial ofG, while the Hosoya index is defined as the total number of the independent edge sets ofG. In this paper, we determine the extremal graph with respect to the matching energy and Hosoya index for all phenylene chains.

The digital function filters – algorithms and applications

2013 ◽  
Vol 61 (2) ◽  
pp. 371-377
Author(s):  
M. Siwczyński ◽  
A. Drwal ◽  
S. Żaba
Keyword(s):  
Digital Filters ◽  
Phase Shifters ◽  
Square Root ◽  
Real Power ◽  
The Real ◽  
Distributed Parameters ◽  
Digital Modeling ◽  
Analysis And Synthesis ◽  
Basic Functions ◽  
Recursive Formulas

Abstract The simple digital filters are not sufficient for digital modeling of systems with distributed parameters. It is necessary to apply more complex digital filters. In this work, a set of filters, called the digital function filters, is proposed. It consists of digital filters, which are obtained from causal and stable filters through some function transformation. In this paper, for several basic functions: exponential, logarithm, square root and the real power of input filter, the recursive algorithms of the digital function filters have been determined The digital function filters of exponential type can be obtained from direct recursive formulas. Whereas, the other function filters, such as the logarithm, the square root and the real power, require using the implicit recursive formulas. Some applications of the digital function filters for the analysis and synthesis of systems with lumped and distributed parameters (a long line, phase shifters, infinite ladder circuits) are given as well.

The Complexity of Model-Checking Tail-Recursive Higher-Order Fixpoint Logic

2021 ◽  
Vol 178 (1-2) ◽  
pp. 1-30
Author(s):  
Florian Bruse ◽  
Martin Lange ◽  
Etienne Lozes
Keyword(s):  
Model Checking ◽  
Lower Bounds ◽  
Expressive Power ◽  
Higher Order ◽  
Order Logic ◽  
High Complexity ◽  
Transition Systems ◽  
Recursive Formulas ◽  
Second Order Logic ◽  
Monadic Second Order Logic

Higher-Order Fixpoint Logic (HFL) is a modal specification language whose expressive power reaches far beyond that of Monadic Second-Order Logic, achieved through an incorporation of a typed λ-calculus into the modal μ-calculus. Its model checking problem on finite transition systems is decidable, albeit of high complexity, namely k-EXPTIME-complete for formulas that use functions of type order at most k < 0. In this paper we present a fragment with a presumably easier model checking problem. We show that so-called tail-recursive formulas of type order k can be model checked in (k − 1)-EXPSPACE, and also give matching lower bounds. This yields generic results for the complexity of bisimulation-invariant non-regular properties, as these can typically be defined in HFL.

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.

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.

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 connection between the Kekulé structures of pentagonal chains and the Hosoya index of caterpillar trees

2017 ◽  
Vol 232 ◽  
pp. 230-234 ◽  
Author(s):  
Chuanqi Xiao ◽  
Haiyan Chen ◽  
Andrei M. Raigorodskii
Keyword(s):  
Hosoya Index ◽  
Kekulé Structures
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