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.

scholarly journals Extremal Matching Energy and the Largest Matching Root of Complete Multipartite Graphs

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-7
Author(s):  
Xiaolin Chen ◽  
Huishu Lian
Keyword(s):  
Split Graph ◽  
Matching Polynomial ◽  
Multipartite Graphs ◽  
Complete Multipartite Graphs ◽  
The Absolute ◽  
Complete Multipartite ◽  
The Given ◽  
Turán Graph

The matching energy ME(G) of a graph G was introduced by Gutman and Wagner, which is defined as the sum of the absolute values of the roots of the matching polynomial m(G,x). The largest matching root λ1(G) is the largest root of the matching polynomial m(G,x). Let Kn1,n2,…,nr denote the complete r-partite graph with order n=n1+n2+…+nr, where r>1. In this paper, we prove that, for the given values n and r, both the matching energy ME(G) and the largest matching root λ1(G) of complete r-partite graphs are minimal for complete split graph CS(n,r-1) and are maximal for Turán graph T(n,r).

scholarly journals The Merrifield-Simmons Index and Hosoya Index ofC(n,k,λ)Graphs

2012 ◽  
Vol 2012 ◽  
pp. 1-8
Author(s):  
Shaojun Dai ◽  
Ruihai Zhang
Keyword(s):  
Hosoya Index ◽  
Independent Vertex ◽  
Vertex Set ◽  
Independent Edge ◽  
Vertex Sets

The Merrifield-Simmons indexi(G)of a graphGis defined as the number of subsets of the vertex set, in which any two vertices are nonadjacent, that is, the number of independent vertex sets ofGThe Hosoya indexz(G)of a graphGis defined as the total number of independent edge subsets, that is, the total number of its matchings. ByC(n,k,λ)we denote the set of graphs withnvertices,kcycles, the length of every cycle isλ, and all the edges not on the cycles are pendant edges which are attached to the same vertex. In this paper, we investigate the Merrifield-Simmons indexi(G)and the Hosoya indexz(G)for a graphGinC(n,k,λ).

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 Linear Algorithms for the Hosoya Index and Hosoya Matrix of a Tree

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 142
Author(s):  
Aleksander Vesel
Keyword(s):  
Linear Time ◽  
Molecular Graph ◽  
Time Algorithm ◽  
Hosoya Index ◽  
Significant Interest ◽  
The Matrix ◽  
Independent Edge ◽  
Linear Algorithms ◽  
Arbitrary Tree ◽  
Structure Descriptor

The Hosoya index of a graph is defined as the total number of its independent edge sets. This index is an important example of topological indices, a molecular-graph based structure descriptor that is of significant interest in combinatorial chemistry. The Hosoya index inspires the introduction of a matrix associated with a molecular acyclic graph called the Hosoya matrix. We propose a simple linear-time algorithm, which does not require pre-processing, to compute the Hosoya index of an arbitrary tree. A similar approach allows us to show that the Hosoya matrix can be computed in constant time per entry of the matrix.

Digraph Energy of Directed Polygons

2020 ◽  
Vol 23 ◽  
Author(s):  
Bo Deng ◽  
Ning Yang ◽  
Weilin Liang ◽  
Xiaoyun Lu
Keyword(s):  
Molecular Orbital ◽  
Adjacency Matrix ◽  
Energy Levels ◽  
Order Relation ◽  
Theoretical Chemistry ◽  
Characteristic Polynomials ◽  
Molecular Graphs ◽  
Conjugated Hydrocarbons ◽  
Quasi Order ◽  
The Absolute

Background: The energy E(G) of G is defined as the sum the absolute values of the eigenvalues of its adjacency matrix. In theoretical chemistry, within the Hu ̈ckel molecular orbital (HMO) approximation, the energy levels of the π-electrons in molecules of conjugated hydrocarbons are related to the energy of the molecular graphs. Objective: Generally, the energy to digraphs was proposed. Methodology: Let Δ_n be the set consisting of digraphs with n vertices and each cycle having length≡2 mod(4). The set of all the n-order directed hollow k-polygons in Δ_n based on a k-polygon G is denoted by H_k (G). Results: In this research, by using the quasi-order relation over Δ_n and the characteristic polynomials of digraphs, we describe the directed hollow k-polygon with the maximum digraph energy in H_k (G). Conclusion: The n-order oriented hollow k-polygon with the maximum digraph energy among Hk(G) only contains a cycle. Moreover, such a cycle is the longest one produced in G.

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 On the tricyclic graphs with three disjoint 6-cycles and maximum matching energy

2015 ◽  
Vol 46 (4) ◽  
pp. 389-399
Author(s):  
Yun-Xia Zhou ◽  
Hong-Hai Li
Keyword(s):  
Maximum Matching ◽  
Matching Polynomial ◽  
Tricyclic Graphs ◽  
The Absolute ◽  
Vertex Disjoint

The matching energy of a graph was introduced recently by Gutman and Wagner and defined as the sum of the absolute values of zeros of its matching polynomial. In this paper, we characterize graphs that attain the maximum matching energy among all connected tricyclic graphs of order $n$ with three vertex-disjoint $C_6$'s.

scholarly journals Hosoya Index ofL-Type Polyphenyl Spiders

2016 ◽  
Vol 2016 ◽  
pp. 1-5
Author(s):  
Ren Shengzhang ◽  
Wu Tingzeng
Keyword(s):  
Hosoya Index ◽  
Independent Edge

The polyphenyl system is composed ofnhexagons obtained from two adjacent hexagons that are sticked by a path with two vertices. The Hosoya index of a graphGis defined as the total number of the independent edge sets ofG. In this paper, we give a computing formula of Hosoya index of a type of polyphenyl system. Furthermore, we characterize the extremal Hosoya index of the type of polyphenyl system.

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.

Bulk standards for low-temperature biological x-ray microanalysis

1984 ◽  
Vol 42 ◽  
pp. 282-283
Author(s):  
P. Echlin ◽  
M. McKoon ◽  
E.S. Taylor ◽  
C.E. Thomas ◽  
K.L. Maloney ◽  
...  
Keyword(s):  
Phase Separation ◽  
Low Temperature ◽  
Analytical Method ◽  
Salt Solutions ◽  
Absolute Concentration ◽  
Cooling Process ◽  
X Ray ◽  
The Absolute ◽  
Analytical Procedures ◽  
Electrical Conductors

Although sections of frozen salt solutions have been used as standards for x-ray microanalysis, such solutions are less useful when analysed in the bulk form. They are poor thermal and electrical conductors and severe phase separation occurs during the cooling process. Following a suggestion by Whitecross et al we have made up a series of salt solutions containing a small amount of graphite to improve the sample conductivity. In addition, we have incorporated a polymer to ensure the formation of microcrystalline ice and a consequent homogenity of salt dispersion within the frozen matrix. The mixtures have been used to standardize the analytical procedures applied to frozen hydrated bulk specimens based on the peak/background analytical method and to measure the absolute concentration of elements in developing roots.

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