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Journal articles on the topic 'Hosoya polynomial'

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Published: 3 June 2025
Last updated: 20 July 2025

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1

Sheikh, Umber, Sidra Rashid, Cenap Ozel, and Richard Pincak. "On Hosoya Polynomial and Subsequent Indices of C4C8(R) and C4C8(S) Nanosheets." Symmetry 14, no. 7 (2022): 1349. http://dx.doi.org/10.3390/sym14071349.

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Chemical structures are mathematically modeled using chemical graphs. The graph invariants including algebraic polynomials and topological indices are related to the topological structure of molecules. Hosoya polynomial is a distance based algebraic polynomial and is a closed form of several distance based topological indices. This article is devoted to compute the Hosoya polynomial of two different atomic configurations (C4C8(R) and C4C8(S)) of C4C8 Carbon Nanosheets. Carbon nanosheets are the most stable, flexible structure of uniform thickness and admit a vast range of applications. The Hosoya polynomial is used to calculate distance based topological indices including Wiener, hyper Wiener and Tratch–Stankevitch–Zafirov Indices. These indices play their part in determining quantitative structure property relationship (QSPR) and quantitative structure activity relationship (QSAR) of chemical structures. The three dimensional presentation of Hosoya polynomial and related distance based indices leads to the result that though the chemical formula for both the sheets is same, yet they possess different Hosoya Polynomials presenting distinct QSPR and QSAR corresponding to their atomic configuration.
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2

Li, Wei, Ailian Chen, and Jiali Wu. "The Edge-Hosoya Polynomial of Catacondensed Benzenoid Graphs Associated with its Hosoya Polynomial." Match Communications in Mathematical and in Computer Chemistry 92, no. 3 (2024): 761–77. http://dx.doi.org/10.46793/match.92-3.761l.

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This paper reveals the relationship between edge-Hosoya polynomial and the Hosoya polynomial of catacondensed benzenoid graphs. The result shows that, for a catacondensed benzenoid graph, the computations of the edge-Hosoya polynomial can be reduced to that of the Hosoya polynomial.
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3

Wu, Tingzeng, Yinggang Bai, and Shoujun Xu. "Extremal Bicyclic Graphs with Respect to Permanental Sums and Hosoya Indices." Axioms 13, no. 5 (2024): 330. http://dx.doi.org/10.3390/axioms13050330.

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Graph polynomials is one of the important research directions in mathematical chemistry. The coefficients of some graph polynomials, such as matching polynomial and permanental polynomial, are related to structural properties of graphs. The Hosoya index of a graph is the sum of the absolute value of all coefficients for the matching polynomial. And the permanental sum of a graph is the sum of the absolute value of all coefficients of the permanental polynomial. In this paper, we characterize the second to sixth minimal Hosoya indices of all bicyclic graphs. Furthermore, using the results, the second to sixth minimal permanental sums of all bicyclic graphs are also characterized.
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4

Knor, Martin, and Niko Tratnik. "A Method for Computing the Edge-Hosoya Polynomial with Application to Phenylenes." match Communications in Mathematical and in Computer Chemistry 89, no. 3 (2023): 605–29. http://dx.doi.org/10.46793/match.89-3.605k.

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The edge-Hosoya polynomial of a graph is the edge version of the famous Hosoya polynomial. Therefore, the edge-Hosoya polynomial counts the number of (unordered) pairs of edges at distance k ≥ 0 in a given graph. It is well known that this polynomial is closely related to the edge-Wiener index and the edge-hyper-Wiener index. As the main result of this paper, we greatly generalize an earlier result by providing a method for calculating the edge-Hosoya polynomial of a graph G which is obtained by identifying two edges of connected bipartite graphs G1 and G2. To show how the main theorem can be used, we apply it to phenylene chains. In particular, we present the recurrence relations and a linear time algorithm for calculating the edge-Hosoya polynomial of any phenylene chain. As a consequence, closed formula for the edge-Hosoya polynomial of linear phenylene chains is derived.
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5

Abbas, Ghulam, Anam Rani, Muhammad Salman, Tahira Noreen, and Usman Ali. "Hosoya properties of the commuting graph associated with the group of symmetries." Main Group Metal Chemistry 44, no. 1 (2021): 173–84. http://dx.doi.org/10.1515/mgmc-2021-0017.

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Abstract A vast amount of information about distance based graph invariants is contained in the Hosoya polynomial. Such an information is helpful to determine well-known distance based molecular descriptors. The Hosoya index or Z-index of a graph G is the total number of its matching. The Hosoya index is a prominent example of topological indices, which are of great interest in combinatorial chemistry, and later on it applies to address several chemical properties in molecular structures. In this article, we investigate Hosoya properties (Hosoya polynomial, reciprocal Hosoya polynomial and Hosoya index) of the commuting graph associated with an algebraic structure developed by the symmetries of regular molecular gones (constructed by atoms with regular atomic-bonding).
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6

Das, Shibsankar, and Shikha Rai. "On the Hosoya polynomial of the third type of the chain hex-derived network." Journal of the Belarusian State University. Mathematics and Informatics, no. 3 (December 23, 2022): 67–78. http://dx.doi.org/10.33581/2520-6508-2022-3-67-78.

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A topological index plays an important role in characterising various physical properties, biological activities, and chemical reactivities of a molecular graph. The Hosoya polynomial is used to evaluate the distance-based topological indices such as the Wiener index, hyper-Wiener index, Harary index, and Tratch – Stankevitch – Zefirov index. In the present study, we determine a closed form of the Hosoya polynomial for the third type of the chain hex-derived network of dimension n and derive the distance-based topological indices of the network with the help of their direct formulas and alternatively via using the obtained Hosoya polynomial. Finally, we graphically represent the computed distance-based topological indices and the Hosoya polynomial of the underlying network to comprehend their geometrical pattern. This study of the Hosoya polynomial and the corresponding indices can set the basis for more exploration into chain hex-derived networks and their properties.
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7

Jabir, Azeez Lafta, AbdulJalil M. Khalaf, and Emad A. Jaffar AL-Mulla. "Hosoya Polynomials Of Some Semiconducotors." Journal of Kufa for Mathematics and Computer 2, no. 2 (2014): 49–55. http://dx.doi.org/10.31642/jokmc/2018/020208.

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The Hosoya polynomial of a graph G is a graphical invariant polynomial that its first derivative at x = 1 is equal to the Wiener index and second derivative at x =1 is equal to the hyperWiener index. In this paper we compute the Hosoya polynomial of some semiconducotors [Caesium Chloride, Perovskite structure, Zinc blende structure, Rock-salt(Nacl)structure, Wurtzite structure, Chalcopyrite structure], Wiener index and hyper-Wiener index for then.The Hosoya polynomial of a graph G is a graphical invariant polynomial that its first derivative at x = 1 is equal to the Wiener index and second derivative at x =1 is equal to the hyperWiener index. In this paper we compute the Hosoya polynomial of some semiconducotors [Caesium Chloride, Perovskite structure, Zinc blende structure, Rock-salt(Nacl)structure, Wurtzite structure, Chalcopyrite structure], Wiener index and hyper-Wiener index for then.
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8

Eliasi, Mehdi, and Bijan Taeri. "Hosoya polynomial of zigzag polyhex nanotorus." Journal of the Serbian Chemical Society 73, no. 3 (2008): 311–19. http://dx.doi.org/10.2298/jsc0803311e.

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The Hosoya polynomial of a molecular graph G is defined as H(G,?)=?{u,v}V?(G) ?d(u,v), where d(u,v) is the distance between vertices u and v. The first derivative of H(G,?) at ?=1 is equal to the Wiener index of G, defined as W(G)?{u,v}?V(G)d(u,v). The second derivative of 1/2 ?H(G, ?) at ?=1 is equal to the hyper-Wiener index, defined as WW(G)+1/2?{u,v}?V(G)d(u,v)?. Xu et al.1 computed the Hosoya polynomial of zigzag open-ended nanotubes. Also Xu and Zhang2 computed the Hosoya polynomial of armchair open-ended nanotubes. In this paper, a new method was implemented to find the Hosoya polynomial of G = HC6[p,q], the zigzag polyhex nanotori and to calculate the Wiener and hyper Wiener indices of G using H(G,?).
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9

Ramane, H. S., A. B. Ganagi, K. P. Narayankar, and S. S. Shirkol. "Terminal Hosoya Polynomial of Line Graphs." Journal of Discrete Mathematics 2013 (June 13, 2013): 1–3. http://dx.doi.org/10.1155/2013/857908.

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The terminal Hosoya polynomial of a graph G is defined as TH(G,λ)=∑k≥1‍dT(G,k)λk, where dT(G,k) is the number of pairs of pendant vertices of G that are at distance k. In this paper we obtain terminal Hosoya polynomial of line graphs.
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10

Gaftan, Awni M., Akram S. Mohammed, and Osama H. Subhi. "Cryptography by Using"Hosoya"Polynomials for"Graphs Groups of Integer Modulen and"Dihedral Groups with'Immersion"Property." Ibn AL- Haitham Journal For Pure and Applied Science 31, no. 3 (2018): 151. http://dx.doi.org/10.30526/31.3.2008.

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In this paper we used Hosoya polynomial ofgroupgraphs Z1,...,Z26 after representing each group as graph and using Dihedral group to"encrypt the plain texts with the immersion property which provided Hosoya polynomial to immerse the cipher text in another"cipher text to become very"difficult to solve.
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11

Shaheen, Ramy, Suhail Mahfud, and Qays Alhawat. "Chromatic Schultz and Gutman Polynomials of Jahangir Graphs J 2 , m and J 3 , m." Journal of Applied Mathematics 2023 (January 4, 2023): 1–13. http://dx.doi.org/10.1155/2023/4891083.

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Topological polynomial and indices based on the distance between the vertices of a connected graph are widely used in the chemistry to establish relation between the structure and the properties of molecules. In a similar way, chromatic versions of certain topological indices and the related polynomial have also been discussed in the recent literature. In this paper, we present the chromatic Schultz and Gutman polynomials and the expanded form of the Hosoya polynomial and chromatic Schultz and Gutman polynomials, and then we derive these polynomials for special cases of Jahangir graphs.
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12

Cheng, Zhong-Lin, Ashaq Ali, Haseeb Ahmad, Asim Naseem, and Maqbool Ahmad Chaudhary. "Hosoya and Harary Polynomials of Hourglass and Rhombic Benzenoid Systems." Journal of Chemistry 2020 (April 9, 2020): 1–14. http://dx.doi.org/10.1155/2020/5398109.

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In the fields of chemical graph theory, topological index is a type of a molecular descriptor that is calculated based on the graph of a chemical compound. In 1947, Harry Wiener introduced “path number” which is now known as Wiener index and is the oldest topological index related to molecular branching. Hosoya polynomial plays a vital role in determining Wiener index. In this report, we compute the Hosoya polynomials for hourglass and rhombic benzenoid systems and recover Wiener and hyper-Wiener indices from them.
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13

Chen, Hanlin, Xueting Liu, and Jinrong Shen. "Matching Polynomials and Independence Polynomials of Benzenoid Chains." Match Communications in Mathematical and in Computer Chemistry 92, no. 3 (2024): 779–809. http://dx.doi.org/10.46793/match.92-3.779c.

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Benzenoid chains, also known as hexagonal chains, are a class of organic compounds that consist of an arrangement of hexagonal rings fused together. In this article, we first present reduction formulas to compute the matching polynomial and independence polynomial of any benzenoid chain by utilizing the transfer matrix technique. Subsequently, computational formulas for the Hosoya index and Merrifield-Simmons index of benzenoid chains are derived. Furthermore, the expected values of the Hosoya index and Merrifield-Simmons index for random benzenoid chains are also obtained.
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14

Stevanović, Dragan. "Hosoya polynomial of composite graphs." Discrete Mathematics 235, no. 1-3 (2001): 237–44. http://dx.doi.org/10.1016/s0012-365x(00)00277-6.

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15

Vlad, M. P., and M. V. Diudea. "Hosoya-Diudea polynomial in hyper structures." Acta Chemica Iasi 21, no. 2 (2013): 83–92. http://dx.doi.org/10.2478/achi-2013-0008.

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Abstract Hosoya polynomial counts finite sequences of distances in a graph G; more exactly, it counts the number of points/atoms lying at a given distance in G. The polynomial coefficients are calculable by means of layer/shell matrices. Shell matrix operator enables the transformation of any square matrix in the corresponding layer/shell matrix, thus generalizing the local property counting according to its distribution by the distances in G. This represents the “Hosoya-Diudea” generalized counting polynomial. We applied this theory to several hypothetical nanostructures with icosahedral symmetry.
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16

Chen, Lian, Abid Mehboob, Haseeb Ahmad, Waqas Nazeer, Muhammad Hussain, and M. Reza Farahani. "Hosoya and Harary Polynomials of TOX(n),RTOX(n),TSL(n) and RTSL(n)." Discrete Dynamics in Nature and Society 2019 (July 16, 2019): 1–18. http://dx.doi.org/10.1155/2019/8696982.

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In the fields of chemical graph theory, topological index is a type of a molecular descriptor that is calculated based on the graph of a chemical compound. In 1947, Wiener introduced “path number” which is now known as Wiener index and is the oldest topological index related to molecular branching. Hosoya polynomial plays a vital role in determining Wiener index. In this report, we computed the Hosoya and the Harary polynomials for TOX(n),RTOX(n),TSL(n), and RTSL(n) networks. Moreover, we computed serval distance based topological indices, for example, Wiener index, Harary index, and multiplicative version of wiener index.
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17

Ali, Fawad, Bilal Ahmad Rather, Anwarud Din, Tareq Saeed, and Asad Ullah. "Power Graphs of Finite Groups Determined by Hosoya Properties." Entropy 24, no. 2 (2022): 213. http://dx.doi.org/10.3390/e24020213.

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Suppose G is a finite group. The power graph represented by P(G) of G is a graph, whose node set is G, and two different elements are adjacent if and only if one is an integral power of the other. The Hosoya polynomial contains much information regarding graph invariants depending on the distance. In this article, we discuss the Hosoya characteristics (the Hosoya polynomial and its reciprocal) of the power graph related to an algebraic structure formed by the symmetries of regular molecular gones. As a consequence, we determined the Hosoya index of the power graphs of the dihedral and the generalized groups. This information is useful in determining the renowned chemical descriptors depending on the distance. The total number of matchings in a graph Γ is known as the Z-index or Hosoya index. The Z-index is a well-known type of topological index, which is popular in combinatorial chemistry and can be used to deal with a variety of chemical characteristics in molecular structures.
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18

Deutsch, Emeric, and Juan Alberto Rodríguez-Velázquez. "The Terminal Hosoya Polynomial of Some Families of Composite Graphs." International Journal of Combinatorics 2014 (April 16, 2014): 1–4. http://dx.doi.org/10.1155/2014/696507.

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Let G be a connected graph and let Ω(G) be the set of pendent vertices of G. The terminal Hosoya polynomial of G is defined as TH(G,t)∶=∑x,y∈Ω(G):x≠ytdG(x,y), where dG(x,y) denotes the distance between the pendent vertices x and y. In this note paper we obtain closed formulae for the terminal Hosoya polynomial of rooted product graphs and corona product graphs.
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19

Araujo, Oswaldo, Mario Estrada, Daniel A. Morales, and Juan Rada. "The higher-order matching polynomial of a graph." International Journal of Mathematics and Mathematical Sciences 2005, no. 10 (2005): 1565–76. http://dx.doi.org/10.1155/ijmms.2005.1565.

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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.
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20

Mohammad, Reza Farahani. "HOSOYA POLYNOMIAL, WIENER AND HYPERWIENER INDICES OF SOME REGULAR GRAPHS." Informatics Engineering, an International Journal (IEIJ) 01, dec (2013): 01–05. https://doi.org/10.5281/zenodo.1435656.

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Let G be a graph. The distance d(u,v) between two vertices u and v of G is equal to the length of a shortest path that connects u and v. The Wiener index W(G) is the sum of all distances between vertices of G, whereas the hyper-Wiener index WW(G) is defined as ( ) ( ) ( ) ( ) ( ) 2 {u,v} V G WW G d v u d v u , , . ∈ = + ∑ Also, the Hosoya polynomial was introduced by H. Hosoya and define ( ) ( ) ( ) , {u,v} V G , . d v u H G x x ∈ = ∑ In this paper, the Hosoya polynomial, Wiener index and Hyper-Wiener index of some regular graphs are determined.
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21

Shaheen, Ramy, Suhail Mahfud, and Qays Alhawat. "Hosoya, Schultz, and Gutman Polynomials of Generalized Petersen Graphs P n , 1 and P n , 2." Journal of Mathematics 2023 (July 3, 2023): 1–18. http://dx.doi.org/10.1155/2023/7341285.

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The graph theory has wide important applications in various other types of sciences. In chemical graph theory, we have many topological polynomials for a graph G through which we can compute many topological indices. Topological indices are numerical values and descriptors which are used to quantify the physiochemical properties and bioactivities of the chemical graph. In this paper, we compute Hosoya polynomial, hyper-Wiener index, Tratch–Stankevitch–Zefirov index, Harary index, Schultz polynomial, Gutman polynomial, Schultz index, and Gutman index of generalized Petersen graphs P n , 1 and P n , 2 .
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22

Movahedi, Fateme. "Matching polynomials for some nanostar dendrimers." Asian-European Journal of Mathematics 14, no. 10 (2021): 2150188. http://dx.doi.org/10.1142/s1793557121501886.

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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.
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23

Xu, Peng, Muhammad Numan, Aamra Nawaz, Saad Ihsan Butt, Adnan Aslam, and Asfand Fahad. "Computing the Hosoya Polynomial of M-th Level Wheel and Its Subdivision Graph." Journal of Chemistry 2021 (November 27, 2021): 1–7. http://dx.doi.org/10.1155/2021/1078792.

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The determination of Hosoya polynomial is the latest scheme, and it provides an excellent and superior role in finding the Weiner and hyper-Wiener index. The application of Weiner index ranges from the introduction of the concept of information theoretic analogues of topological indices to the use as major tool in crystal and polymer studies. In this paper, we will compute the Hosoya polynomial for multiwheel graph and uniform subdivision of multiwheel graph. Furthermore, we will derive two well-known topological indices for the abovementioned graphs, first Weiner index, and second hyper-Wiener index.
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24

Ali, Fawad, Bilal Ahmad Rather, Muhammad Sarfraz, Asad Ullah, Nahid Fatima, and Wali Khan Mashwani. "Certain Topological Indices of Non-Commuting Graphs for Finite Non-Abelian Groups." Molecules 27, no. 18 (2022): 6053. http://dx.doi.org/10.3390/molecules27186053.

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A topological index is a number derived from a molecular structure (i.e., a graph) that represents the fundamental structural characteristics of a suggested molecule. Various topological indices, including the atom-bond connectivity index, the geometric–arithmetic index, and the Randić index, can be utilized to determine various characteristics, such as physicochemical activity, chemical activity, and thermodynamic properties. Meanwhile, the non-commuting graph ΓG of a finite group G is a graph where non-central elements of G are its vertex set, while two different elements are edge connected when they do not commute in G. In this article, we investigate several topological properties of non-commuting graphs of finite groups, such as the Harary index, the harmonic index, the Randić index, reciprocal Wiener index, atomic-bond connectivity index, and the geometric–arithmetic index. In addition, we analyze the Hosoya characteristics, such as the Hosoya polynomial and the reciprocal status Hosoya polynomial of the non-commuting graphs over finite subgroups of SL(2,C). We then calculate the Hosoya index for non-commuting graphs of binary dihedral groups.
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25

Shyama, M. P., and V. Anil Kumar. "On the roots of Hosoya polynomial." Journal of Discrete Mathematical Sciences and Cryptography 19, no. 1 (2016): 199–219. http://dx.doi.org/10.1080/09720529.2015.1117199.

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26

Sadeghieh, Ali, Saeid Alikhani, Nima Ghanbari, Abdul Jalil M. Khalaf, and Hari M. Srivastava. "Hosoya polynomial of some cactus chains." Cogent Mathematics 4, no. 1 (2017): 1305638. http://dx.doi.org/10.1080/23311835.2017.1305638.

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27

Farahani, Mohammad Reza. "Schultz and Modified Schultz Polynomials of Coronene Polycyclic Aromatic Hydrocarbons." International Letters of Chemistry, Physics and Astronomy 32 (April 2014): 1–10. http://dx.doi.org/10.18052/www.scipress.com/ilcpa.32.1.

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Let G = (V;E) be a simple connected graph. The sets of vertices and edges of G are denoted byV = V(G) and E = E(G), respectively. In such a simple molecular graph, vertices represent atoms andedges represent bonds. The distance between the vertices u and v in V(G) of graph G is the number ofedges in a shortest path connecting them, we denote by d(u,v). In graph theory, we have manyinvariant polynomials for a graph G. In this research, we computing the Schultz polynomial, ModifiedSchultz polynomial, Hosoya polynomial and their topological indices of a Hydrocarbon molecule, thatwe call “Coronene Polycyclic Aromatic Hydrocarbons”.
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Farahani, Mohammad Reza. "Schultz and Modified Schultz Polynomials of Coronene Polycyclic Aromatic Hydrocarbons." International Letters of Chemistry, Physics and Astronomy 32 (April 22, 2014): 1–10. http://dx.doi.org/10.56431/p-xn44f2.

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Let G = (V;E) be a simple connected graph. The sets of vertices and edges of G are denoted byV = V(G) and E = E(G), respectively. In such a simple molecular graph, vertices represent atoms andedges represent bonds. The distance between the vertices u and v in V(G) of graph G is the number ofedges in a shortest path connecting them, we denote by d(u,v). In graph theory, we have manyinvariant polynomials for a graph G. In this research, we computing the Schultz polynomial, ModifiedSchultz polynomial, Hosoya polynomial and their topological indices of a Hydrocarbon molecule, thatwe call “Coronene Polycyclic Aromatic Hydrocarbons”.
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29

Rather, Bilal Ahmad, Fawad Ali, Suliman Alsaeed, and Muhammad Naeem. "Hosoya Polynomials of Power Graphs of Certain Finite Groups." Molecules 27, no. 18 (2022): 6081. http://dx.doi.org/10.3390/molecules27186081.

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Assume that G is a finite group. The power graph P(G) of G is a graph in which G is its node set, where two different elements are connected by an edge whenever one of them is a power of the other. A topological index is a number generated from a molecular structure that indicates important structural properties of the proposed molecule. Indeed, it is a numerical quantity connected with the chemical composition that is used to correlate chemical structures with various physical characteristics, chemical reactivity, and biological activity. This information is important for identifying well-known chemical descriptors based on distance dependence. In this paper, we study Hosoya properties, such as the Hosoya polynomial and the reciprocal status Hosoya polynomial of power graphs of various finite cyclic and non-cyclic groups of order pq and pqr, where p,q and r(p≥q≥r) are prime numbers.
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30

Shaheen, Ramy. "The Hosoya polynomial, Wiener index, and Hyper-Wiener index of Jahangir Graph \(J_{8,m}\)." Online Journal of Analytic Combinatorics, no. 15 (December 31, 2020): 1–9. https://doi.org/10.61091/ojac-1515.

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Let \( G(V,E) \) be a simple connected graph with vertex set \( V \) and edge set \( E \). The Wiener index in the graph is \(W(G) = \sum_{\{u,v\} \subseteq V} d(u,v),\) where \( d(u,v) \) is the distance between \( u \) and \( v \), and the Hosoya polynomial of \( G \) is \(H(G, x) = \sum_{\{u,v\} \subseteq V} x^{d(u,v)}.\) The hyper-Wiener index of \( G \) is \(WW(G) = \frac{1}{2} \left( W(G) + \sum_{\{u,v\} \subseteq V} d^2(u,v) \right).\) In this paper, we compute the Wiener index, Hosoya polynomial, and hyper-Wiener index of Jahangir graph \( J_{8,m} \) for \( m \geq 3 \).
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31

Informatics, Engineering an International Journal (IEIJ). "HOSOYA POLYNOMIAL, WIENER AND HYPERWIENER INDICES OF SOME REGULAR GRAPHS." Informatics Engineering, an International Journal (IEIJ) 1, no. 1 (2013): 1–5. https://doi.org/10.5281/zenodo.14382082.

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Let G be a graph. The distance d(u,v) between two vertices u and v of G is equal to the length of a shortest path that connects u and v. The Wiener index W(G) is the sum of all distances between vertices of G,whereas the hyper-Wiener index WW(G) is defined as ( ) ( ) ( ) ( ) ( )2{u,v} V GWW G d v u d v u , , .∈= + ∑ Also, theHosoya polynomial was introduced by H. Hosoya and define ( )( )( ),{u,v} V G, .d v u H G x x∈= ∑ In this paper, the Hosoya polynomial, Wiener index and Hyper Wiener index of some regular graphs are determined.  
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32

Nizami, Abdul Rauf, Asim Naseem, and Hafiz Muhammad Waqar Ahmed. "The polygonal cylinder and its Hosoya polynomial." Online Journal of Analytic Combinatorics, no. 15 (December 31, 2020): 1–12. https://doi.org/10.61091/ojac-1509.

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We introduce a polygonal cylinder \( C_{m,n} \), using the Cartesian product of paths \( P_m \) and \( P_n \) and using topological identification of vertices and edges of two opposite sides of \( P_m \times P_n \), and give its Hosoya polynomial, which, depending on odd and even \( m \), is covered in seven separate cases.
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33

Wu, Tingzeng. "Two Classes of Topological Indices of Phenylene Molecule Graphs." Mathematical Problems in Engineering 2016 (2016): 1–5. http://dx.doi.org/10.1155/2016/8421396.

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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.
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34

Ali, Ahmed, та Haveen Ahmed. "The n-Hosoya Polynomial of 𝑊𝛼⊠ Cβ". AL-Rafidain Journal of Computer Sciences and Mathematics 9, № 2 (2012): 139–50. http://dx.doi.org/10.33899/csmj.2012.163707.

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35

Novak, Tina, Darja Rupnik Poklukar, and Janez Žerovnik. "The Hosoya polynomial of double weighted graphs." Ars Mathematica Contemporanea 15, no. 2 (2018): 441–66. http://dx.doi.org/10.26493/1855-3974.1297.c7c.

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36

Badulin, Dmitry, Alexandr Grebennikov, and Konstantin Vorob'ev. "On the Palindromic Hosoya Polynomial of Trees." MATCH Communications in Mathematical and in Computer Chemistry 88, no. 2 (2022): 471–78. http://dx.doi.org/10.46793/match.88-2.471b.

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37

Xu, Shou-Jun, and Qiu-Xia Zhang. "The Hosoya Polynomial of One-Heptagonal Nanocone." Current Nanoscience 9, no. 3 (2013): 411–14. http://dx.doi.org/10.2174/1573413711309030020.

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38

Xu, Shoujun, and Heping Zhang. "The Hosoya polynomial decomposition for hexagonal chains." Mathematical and Computer Modelling 48, no. 3-4 (2008): 601–9. http://dx.doi.org/10.1016/j.mcm.2007.12.001.

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39

Tratnik, Niko, and Petra Žigert Pleteršek. "The edge-Hosoya polynomial of benzenoid chains." Journal of Mathematical Chemistry 57, no. 1 (2018): 180–89. http://dx.doi.org/10.1007/s10910-018-0942-1.

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40

Deutsch, Emeric, and Juan A. Rodríguez-Velázquez. "The Hosoya polynomial of distance-regular graphs." Discrete Applied Mathematics 178 (December 2014): 153–56. http://dx.doi.org/10.1016/j.dam.2014.06.018.

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41

Liu, Yonghong, Abdul Rauf, AdnanAslam, Saira Ishaq та Abudulai Issa. "Computing Wiener and Hyper-Wiener Indices of Zero-Divisor Graph of ℤ ℊ 3 × ℤ I 1 I 2". Journal of Function Spaces 2022 (19 вересня 2022): 1–11. http://dx.doi.org/10.1155/2022/2046173.

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Let S = ℤ ℊ 3 × ℤ I 1 I 2 be a commutative ring where ℊ , I 1 and I 2 are positive prime integers with I 1 ≠ I 2 . The zero-divisor graph assigned to S is an undirected graph, denoted as Y S with vertex set V( Y (S)) consisting of all Zero-divisor of the ring S and for any c, d ∈ V( Y (S)), c d ∈ E Y S if and only if cd =0. A topological index/descriptor is described as a topological-invariant quantity that transforms a molecular graph into a mathematical real number. In this paper, we have computed distance-based polynomials of Y R i-e Hosoya polynomial, Harary polynomial, and the topological indices related to these polynomials namely Wiener index, and Hyper-Wiener index.
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42

Piyali, Ghosh, Basu Saumya, Karmakar Somnath, and Mandal Bholanath. "Schematic generation of characteristic polynomials and the Hosoya indices of mono- and di-substituted polymer graphs of linear chains and cycles." Journal of Indian Chemical Society Vol. 91, Mar 2014 (2014): 503–15. https://doi.org/10.5281/zenodo.5717684.

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Department of Chemistry, The University of Burdwan, Burdwan-713 104, West Bengal, India <em>E-mail </em>: piyali1979@gmail.com, saumya.basu@rediffmail.com, skarmakarbu@rediffmail.com, bmandal_05@yahoo.com Fax : 91-342-2567938 <em>Manuscript received online 30 March 2013, accepted 17 June 2013</em> The graphs of polymer chains (such as, chain with two pendant vertices at one end and chain with two pendant vertices at each end) and of cycles (such as, cycle with one pendant vertex and cycle with two adjacent pendant vertices) have been considered. Algorithms have been developed for expressing of matching polynomial (MP) coefficients of such graphs as matrix products and subsequently been utilized to express the CP coefficients (in case of monocycles) and the Hosoya indices of such graphs in matrix product forms. Facile computer programs can be made with these algorithms. Melting points (in Kelvin) and motor octane number (MON) have been found to obey linear relationship with the Hosoya indices whereas the boiling points (in Kelvin) bear a excellent linear relationship with the logarithm of Hosoya indices of two kinds of alkanes represented by the linear chains.
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43

Ali, Fawad, Bilal A. Rather, Nahid Fatima, et al. "On the Topological Indices of Commuting Graphs for Finite Non-Abelian Groups." Symmetry 14, no. 6 (2022): 1266. http://dx.doi.org/10.3390/sym14061266.

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A topological index is a number generated from a molecular structure (i.e., a graph) that indicates the essential structural properties of the proposed molecule. Indeed, it is an algebraic quantity connected with the chemical structure that correlates it with various physical characteristics. It is possible to determine several different properties, such as chemical activity, thermodynamic properties, physicochemical activity, and biological activity, using several topological indices, such as the geometric-arithmetic index, arithmetic-geometric index, Randić index, and the atom-bond connectivity indices. Consider G as a group and H as a non-empty subset of G. The commuting graph C(G,H), has H as the vertex set, where h1,h2∈H are edge connected whenever h1 and h2 commute in G. This article examines the topological characteristics of commuting graphs having an algebraic structure by computing their atomic-bond connectivity index, the Wiener index and its reciprocal, the harmonic index, geometric-arithmetic index, Randić index, Harary index, and the Schultz molecular topological index. Moreover, we study the Hosoya properties, such as the Hosoya polynomial and the reciprocal statuses of the Hosoya polynomial of the commuting graphs of finite subgroups of SL(2,C). Finally, we compute the Z-index of the commuting graphs of the binary dihedral groups.
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44

Poklukar, Darja Rupnik, and Janez Žerovnik. "Reliability Hosoya-Wiener Polynomial of Double Weighted Trees*." Fundamenta Informaticae 147, no. 4 (2016): 447–56. http://dx.doi.org/10.3233/fi-2016-1416.

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45

Xu, Shoujun, and Heping Zhang. "The Hosoya polynomial decomposition for catacondensed benzenoid graphs." Discrete Applied Mathematics 156, no. 15 (2008): 2930–38. http://dx.doi.org/10.1016/j.dam.2007.12.004.

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46

Zhang, Qiu-Xia, Shou-Jun Xu, and Hai-Yang Chen. "The Hosoya Polynomial of One-Pentagonal Carbon Nanocone." Fullerenes, Nanotubes and Carbon Nanostructures 22, no. 10 (2014): 866–73. http://dx.doi.org/10.1080/1536383x.2013.812634.

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47

Shiralashetti, S. C., H. S. Ramane, R. A. Mundewadi, and R. B. Jummannaver. "A Comparative Study on Haar Wavelet and Hosaya Polynomial for the numerical solution of Fredholm integral equations." Applied Mathematics and Nonlinear Sciences 3, no. 2 (2018): 447–58. http://dx.doi.org/10.21042/amns.2018.2.00035.

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AbstractIn this paper, a comparative study on Haar wavelet method (HWM) and Hosoya Polynomial method(HPM) for the numerical solution of Fredholm integral equations. Illustrative examples are tested through the error analysis for efficiency. Numerical results are shown in the tables and figures.
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48

Farahani, Mohammad Reza. "ON THE SCHULTZ POLYNOMIAL AND HOSOYA POLYNOMIAL OF CIRCUMCORONENE SERIES OF BENZENOID." Journal of applied mathematics & informatics 31, no. 5_6 (2013): 595–608. http://dx.doi.org/10.14317/jami.2013.595.

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49

Nizami, Abdul Rauf, Muhammad Idrees, Numan Amin, and Zaffar Iqbal. "Hosoya polynomial and topological indices of n-linear benzene." Journal of the National Science Foundation of Sri Lanka 47, no. 2 (2019): 169. http://dx.doi.org/10.4038/jnsfsr.v47i2.9152.

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50

Ahmad, Ali, and S. C. López. "Distance-Based Topological Polynomials Associated with Zero-Divisor Graphs." Mathematical Problems in Engineering 2021 (May 27, 2021): 1–8. http://dx.doi.org/10.1155/2021/4959559.

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Let R be a commutative ring with nonzero identity and let Z R be its set of zero divisors. The zero-divisor graph of R is the graph Γ R with vertex set V Γ R = Z R ∗ , where Z R ∗ = Z R \ 0 , and edge set E Γ R = x , y : x ⋅ y = 0 . One of the basic results for these graphs is that Γ R is connected with diameter less than or equal to 3. In this paper, we obtain a few distance-based topological polynomials and indices of zero-divisor graph when the commutative ring is ℤ p 2 q 2 , namely, the Wiener index, the Hosoya polynomial, and the Shultz and the modified Shultz indices and polynomials.
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