Hosoya Index Research Articles

An efficient Hosoya index algorithm and its application

The number of matching's in a graph is known as the Hosoya index of the graph. The problem of computing Hosoya index is #P-complete. If the adjacent edges are sequentially ordered, then we show that a polynomial algorithm can be designed. The significance of this algorithm is demonstrated by computing Hosoya index for certain chemical compounds such as Pyroxene. This algorithm can be applied to grid like chemical compounds such as sodium chloride, carbon nanotubes, naphtalenic nanotube etc.

The Hosoya index and the Merrifield\u2013Simmons index

In this article, we give sharp bounds on the Hosoya index and the Merrifield–Simmons index for connected graphs of fixed size. As a consequence, we determine all connected graphs of any fixed order and size which maximize the Merrifield–Simmons index. Sharp lower bounds on the Hosoya index are known for graphs of order n and size min [n-1,2n-3]cup left( {n-1atopwithdelims ()2},{natopwithdelims ()2}right] ; while sharp upper bounds were only known for graphs of order n and size mle n+2. We give sharp upper bounds on the Hosoya index for dense graphs with mge {natopwithdelims ()2}-2n/3. Moreover, all extreme graphs are also determined.

Hosoya Index of Two Types of Four Leaves Polyphenyl Systems*

ABSTRACTThe polyphenyl system is composed of n hexagons, where consecutive hexagons are sticked by a path with two vertices. The Hosoya index of a graph G is defined as the total number of the independent edge sets of G. In this paper, we give two computation formulas of Hosoya index of two types of four leaves polyphenyl systems. In particular, we characterize the extremal Hosoya index of two types of four leaves polyphenyl systems.

Extremal bipartite graphs of given connectivity with respect to matching energy

The matching energy of a graph G is ME(G)=2π∫0∞1x2ln[∑k≥0m(G,k)x2k]dx, and the Hosoya index of G is Z(G)=∑k≥0m(G,k), where m(G,k) is the number of k-matchings in G. In this note, we first determine the maximum values of m(G,k) in all connected bipartite graphs with n vertices and a given connectivity. And then we determine the maximum matching energy (resp. Hosoya index) among all connected bipartite graphs with n vertices and a given (edge) connectivity and characterize the corresponding extremal graphs.

Hosoya and Merrifield-Simmons indices in random polyphenyl chains

The Hosoya index of a graph is the total number of matchings in it. And the Merrifield-Simmons index is the total number of independent sets in it. They are typical examples of graph invariants used in mathematical chemistry for quantifying relevant details of molecular structure. In this paper, we obtain explicit analytical expressions for the expectations of the Hosoya index and the Merrifield-Simmons index of a random polyphenyl chain.

Link between Hosoya index and Fibonacci numbers

Link between Hosoya index and Fibonacci numbers

Matching polynomial coefficients and the Hosoya indices of poly(p-phenylene) graphs

ABSTRACTA general approach to determine the matching polynomial (MP) of a graph with two parts connected by an edge is presented in matrix product that is ultimately used in deducing recursion formulas for obtaining the MP coefficients of linear and cylindrical poly(p-phenylene) (PPP) graphs. The Hosoya indices of linear and cylindrical PPPs are derived in terms of that of the two immediately preceding graphs as well as in analytical forms with the use of transfer matrices. Ambient condition density and bulk modulus of linear PPPs with 2–6 phenyl rings have been found to correlate well with the logarithm of their Hosoya indices. Excellent correlations of diameters with the logarithm of Hosoya indices and strain energies with the inverse of the logarithm of Hosoya indices for cylindrical PPPr with r (= 6–16, 18, 20) phenyl rings are obtained. The linear relation between the logarithm of Hosoya indices and diameter and the inverse relation between diameter and strain energy corroborate the fact.

Lucas Sequences and the Hosoya Index of Graphs

Lucas Sequences and the Hosoya Index of Graphs

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

Let Pn be a pentagonal chain. Motivated by the work of Gutman (1977), this paper shows that for a hexagonal chain H, there exists a caterpillar tree T(H) such that the number of Kekulé structures of H is equal to the Hosoya index of T(H). In this paper, we show that for a pentagonal chain Pn with even number of pentagons, there exists a caterpillar tree Tn2 such that the number of Kekulé structures of Pn is equal to the Hosoya index of Tn2. This result can be generalized to any polygonal chain Qn with even number of odd polygons.

Quantitative Structure–Retention Relationship Modeling of Morphine and Its Derivatives on OV-1 Column in Gas–Liquid Chromatography Using Genetic Algorithm

In this paper, a back propagation artificial neural network (BP-ANN) was used for quantitative structure–retention relationship (QSRR) modeling of retention time (t R) of 57 morphine and its derivatives. The molecular descriptors were calculated for each compound. By applying a genetic algorithm, the most relevant descriptors were selected to build the QSRR model. The selected descriptors were: Hosoya Index, kappa1, and most negative potential. The prediction results from the BP-ANN were in good agreement with the experimental values. The optimal QSRR model was developed based on a 3-3-1 artificial neural network architecture using molecular descriptors calculated from molecular structure alone. The root-mean-square error (RMSE) and squared correlation coefficient (R 2) for the ANN model were 0.3996 and 0.9559 for the training set (42 molecules) and 0.6052 and 0.9540 for the prediction set (15 molecules), respectively.

Determining Lucas identities by using Hosoya index

We introduce a new identity of Lucas number by using the Hosoya index. As a consequence we give some properties of Lucas numbers and the extension of the work of Hillard and Windfeldt.

On the Hosoya index of a family of deterministic recursive trees

Abstract In this paper, we calculate the Hosoya index in a family of deterministic recursive trees with a special feature that includes new nodes which are connected to existing nodes with a certain rule. We then obtain a recursive solution of the Hosoya index based on the operations of a determinant. The computational complexity of our proposed algorithm is O ( log 2 n ) with n being the network size, which is lower than that of the existing numerical methods. Finally, we give a weighted tree shrinking method as a graphical interpretation of the recurrence formula for the Hosoya index.

Extremal cacti of given matching number with respect to the distance spectral radius

A cactus is a connected graph in which any two cycles have at most one common vertex. The distance spectral radius ρ(G) of a graph G is the largest eigenvalue of the distance matrix D(G). Recently, many researchers proposed the use of ρ(G) as a molecular structure descriptor of alkanes. In this paper, we characterize n-vertex cyclic cactus with given matching number m which minimizes the distance spectral radius. The resulting cactus also minimizes the Hosoya index, the Wiener index and the Randić index in the same class of graphs.

Abstract P5-07-12: Local nuclear architecture features from H&E images predict early versus distant recurrence in lymph node negative, ER+ breast cancers

Abstract Introduction: Breast cancer (BCa) Patients with ER+ tumors that are lymph node negative (LN-) typically receive hormonal therapy. There is a need to identify ER+ LN- patients that will not benefit from adjuvant chemotherapy and will respond to hormonal therapy alone. Oncotype DX, a quantitative prognostic and predictive gene assay, provides a recurrence score that has been correlated with distant and early recurrence. In this work we present an approach that employ computer extracted features of nuclear architecture and morphology from routine H&E slides alone that can distinguish early and distant recurrence in ER+ breast cancers. By constructing graph networks within epithelium and stroma regions, built using nuclei as vertices and edge connections between proximal nuclei, local nuclear architecture can be quantitatively characterized. Hosoya index (HI) (originally introduced for analysis of chemical bonds) is a measure of a bond (in this context nuclei connections in a graph). In this work, we leverage HI to measure structural similarities of graphs across the populations that are indicative of recurrence in LN- ER+ breast cancer tissue microarray (TMA) images. Design: In this study we considered two tissue microarrays (TMAs) comprising 453 early-stage lymph-node negative (LN-) estrogen receptor positive (ER+) breast cancer (BCa) patients (diagnosed with invasive ductal carcinoma), with a total of N=90 patients experiencing lifetime distant recurrence and N=343 patients who did not. All TMA cores were digitized at 20x magnification (0.33 um/pixel spatial resolution) using a digital whole-slide scanner. Each nucleus was identified via an automated computerized image analysis algorithm developed by our group. Then, using a cluster cell graph that encodes a link between a pair of nodes based on proximity, a series of graphs are constructed for a TMA. A HI value was then assigned to each local graph. A support vector machine classifier was trained in conjunction with the distribution of HI values for the early and distant recurrence cases on the training TMA (n=243, 50 early recurrences). Independent validation of the SVM classifier was performed on the second TMA (n=210, 40 early recurrences). Results: For the LN- ER+ breast cancer dataset, our method was able to distinguish tumors with early and distant recurrence with an accuracy of 75.4%, a positive predictive value of 78.6% and a negative predictive value of 76.4%. The separation between the Kaplan-Meier curves for early and distant recurrence of LN-, ER+ breast cancers on the validation set was statistically significant (p < 0.00102). Conclusion: Based only on tiny H&E punches, a computer-aided morphometric classifier appears to identify lymph node negative, ER+ breast cancers with a low likelihood of recurrence. With further validation, this approach could be developed into an image based assay which could serve as a lower cost alternative to Oncotype DX. Citation Format: Ali S, Rimm D, Ganesan S, Madabhushi A. Local nuclear architecture features from H&E images predict early versus distant recurrence in lymph node negative, ER+ breast cancers. [abstract]. In: Proceedings of the Thirty-Eighth Annual CTRC-AACR San Antonio Breast Cancer Symposium: 2015 Dec 8-12; San Antonio, TX. Philadelphia (PA): AACR; Cancer Res 2016;76(4 Suppl):Abstract nr P5-07-12.

On the Wiener polarity index of graphs

The Wiener polarity index Wp(G) of a graph G is the number of unordered pairs of vertices {u, v} in G such that the distance between u and v is equal to 3. Very recently, Zhang and Hu studied the Wiener polarity index in [Y. Zhang, Y. Hu, 2016] [38]. In this short paper, we establish an upper bound on the Wiener polarity index in terms of Hosoya index and characterize the corresponding extremal graphs. Moreover, we obtain Nordhaus–Gaddum-type results for Wp(G). Our lower bound on Wp(G)+Wp(G¯) is always better than the previous lower bound given by Zhang and Hu.

The largest n - 1 Hosoya indices of unicyclic graphs

The Hosoya index Z(G) of a graph G is defined as the total number of edge independent sets of G. In this paper, we extend the research of [J. Ou, On extremal unicyclic molecular graphs with maximal Hosoya index, Discrete Appl. Math. 157 (2009) 391-397.] and [Y. Ye, X. Pan, H. Liu, Ordering unicyclic graphs with respect to Hosoya indices and Merrifield-Simmons indices, MATCH Commun. Math. Comput. Chem. 59 (2008) 191-202.] and order the largest n - 1 unicyclic graphs with respect to the Hosoya index.

Two Classes of Topological Indices of Phenylene Molecule Graphs

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.

Hosoya Index ofL-Type Polyphenyl Spiders

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.

Merrifield-Simmons and Hosoya index of thorn-regular graphs

This paper deals with the Merrifield-Simmons and Hosoya indices of some thorn graphs. We outline a method for the calculation of these indices in the case of regular caterpillars and regular cyclic caterpillars. Then, by using a result of Belbachir & Bencherif, [Linear recurrent sequences and powers of a square matrix, Integers, Vol. 6, 2006, #A12], we obtain combinatorial expressions for these indices.

Kekulé structures of square–hexagonal chains and the Hosoya index of caterpillar trees

Let Rn be a square–hexagonal chain. In this paper, we show that there exists a caterpillar tree Tn such that the number of Kekulé structures of Rn is equal to the Hosoya index of Tn. Since both hexagonal chains and polyomino chains can be viewed as special square–hexagonal chains, our result generalizes the corresponding results for hexagonal chains (Gutman, 1977) and polyomino chains (Liand Yan, 2012).