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Journal articles on the topic 'Wiener index'

Author: Grafiati

Published: 3 June 2025

Last updated: 31 July 2025

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1

Rajendra, R., P. Siva Kota Reddy, and M. Prabhavathi. "COMPUTATION OF WIENER INDEX, RECIPROCAL WIENER INDEX AND PERIPHERAL WIENER INDEX USING ADJACENCY MATRIX." South East Asian J. of Mathematics and Mathematical Sciences 18, no. 03 (2022): 275–82. http://dx.doi.org/10.56827/seajmms.2022.1803.23.

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2

Ilić, Aleksandar, and Milovan Ilić. "Generalizations of Wiener Polarity Index and Terminal Wiener Index." Graphs and Combinatorics 29, no. 5 (2012): 1403–16. http://dx.doi.org/10.1007/s00373-012-1215-6.

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3

Adnan, Mr, Syed Ahtsham Ul Haq Bokhary, and Muhammad Imran. "On Wiener Polarity Index and Wiener Index of Certain Triangular Networks." Journal of Chemistry 2021 (December 20, 2021): 1–20. http://dx.doi.org/10.1155/2021/2757925.

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A topological index of graph G is a numerical quantity which describes its topology. If it is applied to the molecular structure of chemical compounds, it reflects the theoretical properties of the chemical compounds. A number of topological indices have been introduced so far by different researchers. The Wiener index is one of the oldest molecular topological indices defined by Wiener. The Wiener index number reflects the index boiling points of alkane molecules. Quantitative structure activity relationships (QSAR) showed that they also describe other quantities including the parameters of i

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4

Yu, Gui-dong, Li-fang Ren, and Xing-xing Li. "Wiener Index, Hyper-Wiener Index, Harary Index and Hamiltonicity Properties of graphs." Applied Mathematics-A Journal of Chinese Universities 34, no. 2 (2019): 162–72. http://dx.doi.org/10.1007/s11766-019-3565-9.

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5

Iranmanesh, Ali, Y. Alizadeh, and S. Mirzaie. "Computing Wiener Polynomial, Wiener Index and Hyper Wiener Index of C80Fullerene by GAP Program." Fullerenes, Nanotubes and Carbon Nanostructures 17, no. 5 (2009): 560–66. http://dx.doi.org/10.1080/15363830903133204.

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6

Gutman, Ivan, Boris Furtula, and Miroslav Petrović. "Terminal Wiener index." Journal of Mathematical Chemistry 46, no. 2 (2008): 522–31. http://dx.doi.org/10.1007/s10910-008-9476-2.

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7

Nikolić, Sonja, Nenad Trinajstić, and Milan Randić. "Wiener index revisited." Chemical Physics Letters 333, no. 3-4 (2001): 319–21. http://dx.doi.org/10.1016/s0009-2614(00)01367-1.

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8

Li, X. H. "The extended hyper-Wiener index." Canadian Journal of Chemistry 81, no. 9 (2003): 992–96. http://dx.doi.org/10.1139/v03-106.

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According to the definition of molecular connectivity and the definition of a hyper-Wiener index, a novel set of hyper-Wiener indexes (Rn, mRn) are defined and are named the extended hyper-Wiener indexes. Where n = 1, 2, 3, 4,... represents the type of subgraph units and is the number of endmost atoms of the subgraph unit, m is the number of atoms of the subgraph unit. Here n = 1 means the subgraph unit is an atom, n = 2 means the subgraph units are straight-line combinations of m atoms (m = 2, 3, 4, 5, 6,...), and n = 3 means the subgraph units are Y types of combinations of m atoms (m = 4, 5

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9

Heydari, Abbas. "On the Wiener index and the hyper-Wiener index of the Kragujevac trees." Proyecciones (Antofagasta) 42, no. 3 (2023): 727–39. http://dx.doi.org/10.22199/issn.0717-6279-4804.

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In this paper, the Wiener index and the hyper-Wiener index of the Kragujevac trees is computed in term of its vertex degrees. As application, we obtain an upper bond and a lower bound for the Wiener index and the hyper-Wiener index of these trees.

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10

Gutman, Ivan. "Relation between hyper-Wiener and Wiener index." Chemical Physics Letters 364, no. 3-4 (2002): 352–56. http://dx.doi.org/10.1016/s0009-2614(02)01343-x.

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11

Ahmad, Sarfraz, Hafiz Muhammad Afzal Siddiqui, Arfan Ali, Mohammad R. Farahani, Muhammad Imran, and Ismail Naci Cangul. "On Wiener index and Wiener polarity index of some polyomino chains." Journal of Discrete Mathematical Sciences and Cryptography 22, no. 7 (2019): 1151–64. http://dx.doi.org/10.1080/09720529.2019.1688965.

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12

Gutman, Ivan, and Boris Furtula. "Hyper-Wiener Index vs. Wiener Index. Two Highly Correlated Structure-Descriptors." Monatshefte für Chemie - Chemical Monthly 134, no. 7 (2003): 975–81. http://dx.doi.org/10.1007/s00706-003-0003-7.

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13

高, 云. "Some Conclusions on Modified Wiener Index and Modified Hyper-Wiener Index." Biophysics 03, no. 03 (2015): 59–66. http://dx.doi.org/10.12677/biphy.2015.33006.

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14

Cavaleri, Matteo, Daniele D’Angeli, Alfredo Donno, and Stefan Hammer. "Wiener, edge-Wiener, and vertex-edge-Wiener index of Basilica graphs." Discrete Applied Mathematics 307 (January 2022): 32–49. http://dx.doi.org/10.1016/j.dam.2021.09.025.

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15

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 hyperï€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

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16

Johar, Dwindi Agryanti, Asep Kuswandi Supriatna, and Ema Carnia. "Wiener Index Calculation on the Benzenoid System: A Review Article." Jurnal Matematika Integratif 19, no. 1 (2023): 13. http://dx.doi.org/10.24198/jmi.v19.n1.44487.13-28.

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The Weiner index is considered one of the basic descriptors of fixed interconnection networks because it provides the average distance between any two nodes of the network. Many methods have been used by researchers to calculate the value of the Wiener index. starting from the brute force method to the invention of an algorithm to calculate the Wiener index without calculating the distance matrix. The application of the Wiener index is found in the molecular structure of organic compounds, especially the benzenoid system. The value of the Wiener index of a molecule is closely related to its ph

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17

Johar, Dwindi Agryanti, Asep Kuswandi Supriatna, and Ema Carnia. "Wiener Index Calculation on the Benzenoid System: A Review Article." Jurnal Matematika Integratif 19, no. 1 (2023): 13. http://dx.doi.org/10.24198/jmi.v19.n1.44487.13-30.

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The Weiner index is considered one of the basic descriptors of fixed interconnection networks because it provides the average distance between any two nodes of the network. Many methods have been used by researchers to calculate the value of the Wiener index. starting from the brute force method to the invention of an algorithm to calculate the Wiener index without calculating the distance matrix. The application of the Wiener index is found in the molecular structure of organic compounds, especially the benzenoid system. The value of the Wiener index of a molecule is closely related to its ph

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18

Babujee, J. Baskar, and J. Senbagamalar. "On Wiener and terminal Wiener index of graphs." International Journal of Biomathematics 08, no. 05 (2015): 1550066. http://dx.doi.org/10.1142/s1793524515500667.

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The Wiener index is a topological index defined as the sum of distances between all pairs of vertices in a graph. It was introduced as a structural descriptor for molecular graphs of alkanes, which are trees with vertex degrees of four at the most. The terminal Wiener index is defined as the sum of distances between all pairs of pendent vertices in a graph. In this paper we investigate Wiener and terminal Wiener for graphs derived from certain operations.

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19

Eliasi, Mehdi, and Bijan Taeri. "Extension of the Wiener index and Wiener polynomial." Applied Mathematics Letters 21, no. 9 (2008): 916–21. http://dx.doi.org/10.1016/j.aml.2007.10.001.

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20

V. KALADEVI, R. BHUVANESHWARI,. "NORDHAUS – GADDUM TYPE RESULTS FOR WIENER LIKE INDICES OF GRAPHS." Psychology and Education Journal 58, no. 2 (2021): 5843–54. http://dx.doi.org/10.17762/pae.v58i2.3042.

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A Nordhaus - Gaddum type result is a lower or upper bound on sum or product of a parameter of a graph and its complement. This concept was introduced in 1956 by Nordhaus E. A., Gaddum J. W. Generalized Wiener like indices such as wiener index, Detour index, Reciprocal- wiener index, Harary- wiener index, Hyper- wiener index, Reciprocal- Detour index, Harary- Detour index and Hyper- Detour index have been studied in graph theory. In this paper, Nordhaus – Gaddum type results of these indices for k-Sun graph and four regular graph are presented.

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21

Lin, Zhen, and Ting Zhou. "Degree-weighted Wiener index of a graph." Mathematical Modelling and Control 4, no. 1 (2024): 9–16. http://dx.doi.org/10.3934/mmc.2024002.

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<abstract><p>From geometric point of view, we introduced the Sombor-Wiener index of a graph and studied the basic properties of the new index. It was shown that the Sombor-Wiener index was useful in predicting the acentric factor of octane isomers. In addition, we proposed a degree-weighted Wiener index to generalize the Schultz index, the Gutman index, and the Sombor-Wiener index. Meanwhile, we gave the calculation formula of degree-weighted Wiener index for generalized Bethe trees.</p></abstract>

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22

Shi, Xiaolong, Maryam Akhoundi, Ali Asghar Talebi, and Seyed Hossein Sadati. "Some Properties of Cubic Fuzzy Graphs with an Application." Symmetry 14, no. 12 (2022): 2623. http://dx.doi.org/10.3390/sym14122623.

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The advent of fuzzy sets, and consequently fuzzy graphs, has solved many problems in ambiguous and uncertain contexts. It is interesting and necessary to study the Wiener index in a cubic fuzzy graph that employs both fuzzy membership and interval-valued fuzzy membership at the same time. In this paper, the Wiener index in a cubic fuzzy graph is introduced as a cubic fuzzy number and some related results are described. The comparison between connectivity index and Wiener index, changes in Wiener index through deleting a node or an edge, and determining the Wiener index in some specific cubic f

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23

Alhevaz, Abdollah, Maryam Baghipur, and Sadegh Rahimi. "Bounds on hyper-Wiener index of graphs." Asian-European Journal of Mathematics 10, no. 03 (2017): 1750057. http://dx.doi.org/10.1142/s1793557117500577.

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The Wiener number [Formula: see text] of a graph [Formula: see text] was introduced by Harold Wiener in connection with the modeling of various physic-chemical, biological and pharmacological properties of organic molecules in chemistry. Milan Randić introduced a modification of the Wiener index for trees (acyclic graphs), and it is known as the hyper-Wiener index. Then Klein et al. generalized Randić’s definition for all connected (cyclic) graphs, as a generalization of the Wiener index, denoted by [Formula: see text] and defined as [Formula: see text]. In this paper, we establish some upper

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24

Liu, Jiangyi. "Sufficient Conditions for Wiener Index, Hyper-Wiener Index and Harary Index of the Hamilton Graph." Mathematics and Computer Science 10, no. 1 (2025): 15–18. https://doi.org/10.11648/j.mcs.20251001.12.

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The topological index can be used to depict the structural properties of graphs, and the Hamiltonian problem of graphs has always been a classical problem in graph theory. In this work, we use some known conditions to give some sufficient conditions for Hamilton graphs by the Wiener index, Hyper-Wiener index and Harary index of a graph.

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25

Li, X. H. "The extended Wiener index." Chemical Physics Letters 365, no. 1-2 (2002): 135–39. http://dx.doi.org/10.1016/s0009-2614(02)01408-2.

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26

Yun-ping, Huang, Wang Zhen-dong, Yang Feng, and Zhou Pei-jiang. "Reforming of wiener index." Wuhan University Journal of Natural Sciences 9, no. 1 (2004): 115–19. http://dx.doi.org/10.1007/bf02912731.

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27

Castro, Eduardo, Ivan Gutman, Damian Marino, and Pablo Peruzzo. "Upgrading the Wiener index." Journal of the Serbian Chemical Society 67, no. 10 (2002): 647–51. http://dx.doi.org/10.2298/jsc0210647c.

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The Wiener index Wis the oldest molecular-graph-based structure-descriptor. It is defined as the sum of the distances of all pairs of vertices of the molecular graph G, where the distance is the number of edges in the shortest path connecting the respective vertices, and where G is the hydrogen-depleted molecular graph. This seemingly very simple topological index could be "upgraded" (a) by using as the distance the sum of the bond lengths along the shortest path, or (b) by using the Euclidean distance between the respective pairs of atoms. Each of these "upgraded" Wiener indices could be comp

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28

Pleteršek, Petra Žigert. "The edge-Wiener index and the edge-hyper-Wiener index of phenylenes." Discrete Applied Mathematics 255 (February 2019): 326–33. http://dx.doi.org/10.1016/j.dam.2018.07.024.

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29

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 multiplic

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30

Sulphikar, A. "Terminal Wiener Index of Fibonacci trees." International Journal of Engineering and Advanced Technology (IJEAT) 9, no. 3 (2020): 2533–35. https://doi.org/10.35940/ijeat.B4542.029320.

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The terminal Wiener index of a tree is defined as the sum of distances between all leaf pairs of T. We derive closed form expression for the terminal Wiener index of fibonacci trees. We also describe a linear time algorithm to compute terminal Wiener index of a tree.

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31

Azari, M., and A. Iranmanesh. "On the edge-Wiener index of the disjunctive product of simple graphs." Algebra and Discrete Mathematics 30, no. 1 (2020): 1–14. http://dx.doi.org/10.12958/adm242.

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The edge-Wiener index of a simple connected graph G is defined as the sum of distances between all pairs of edges of G where the distance between two edges in G is the distance between the corresponding vertices in the line graph of G. In this paper, we study the edge-Wiener index under the disjunctive product of graphs and apply our results to compute the edge-Wiener index for the disjunctive product of paths and cycles.

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32

Soltani, Abolghasem, and Ali Iranmanesh. "On the Edge Wiener index." Filomat 28, no. 3 (2014): 541–49. http://dx.doi.org/10.2298/fil1403541s.

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Let G be a simple connected graph. The Wiener index of G is the sum of all distances between vertices of G. Whereas, the edge Wiener index of G is defined as the sum of distances between all pairs of edges of G where the distance between the edges f and g in E(G) is defined as the distance between the vertices f and g in the line graph of G. In this paper we will describe a new method for calculating the edge Wiener index. Then find this index for the triangular graphs. Also, we obtain an explicit formula for the Wiener index of the Cartesian product of two graphs using the group automorphisms

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33

Zhang, Jie, Guang-Jun Zhang, Hua Wang, and Xiao-Dong Zhang. "Extremal trees with respect to the Steiner Wiener index." Discrete Mathematics, Algorithms and Applications 11, no. 06 (2019): 1950067. http://dx.doi.org/10.1142/s1793830919500678.

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The well-known Wiener index is defined as the sum of pairwise distances between vertices. Extremal problems with respect to it have been extensively studied for trees. A generalization of the Wiener index, called the Steiner Wiener index, takes the sum of minimum sizes of subgraphs that span [Formula: see text] given vertices over all possible choices of the [Formula: see text] vertices. We consider the extremal problems with respect to the Steiner Wiener index among trees of a given degree sequence. First, it is pointed out minimizing the Steiner Wiener index in general may be a difficult pro

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34

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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35

Xing, Rundan, and Bo Zhou. "Ordering trees having small reverse wiener indices." Filomat 26, no. 4 (2012): 637–48. http://dx.doi.org/10.2298/fil1204637x.

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The Wiener index W(G) of a connected graph G is defined as the sum of distances between all unordered pairs of vertices of G. As a variation of the Wiener index, the reverse Wiener index of G is defined as ?(G) = ? n(n ? 1)d ? W(G), where n is the number of vertices, and d is the diameter of G. It is known that the star is the unique n-vertex tree with the smallest reverse Wiener index. We now determine the second and the third smallest reverse Wiener indices of n-vertex trees, and characterize the trees whose reverse Wiener indices attain these values for n ? 5.

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36

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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37

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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38

Bi, Bo, Muhammad Kamran Jamil, Khawaja Muhammad Fahd, Tian-Le Sun, Imran Ahmad, and Lei Ding. "Algorithms for Computing Wiener Indices of Acyclic and Unicyclic Graphs." Complexity 2021 (May 3, 2021): 1–6. http://dx.doi.org/10.1155/2021/6663306.

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Let G = V G , E G be a molecular graph, where V G and E G are the sets of vertices (atoms) and edges (bonds). A topological index of a molecular graph is a numerical quantity which helps to predict the chemical/physical properties of the molecules. The Wiener, Wiener polarity, and the terminal Wiener indices are the distance-based topological indices. In this paper, we described a linear time algorithm (LTA) that computes the Wiener index for acyclic graphs and extended this algorithm for unicyclic graphs. The same algorithms are modified to compute the terminal Wiener index and the Wiener pol

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39

Gao, Tingmei, and Iftikhar Ahmed. "Distance-Based Polynomials and Topological Indices for Hierarchical Hypercube Networks." Journal of Mathematics 2021 (September 11, 2021): 1–11. http://dx.doi.org/10.1155/2021/5877593.

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Topological indices are the numbers associated with the graphs of chemical compounds/networks that help us to understand their properties. The aim of this paper is to compute topological indices for the hierarchical hypercube networks. We computed Hosoya polynomials, Harary polynomials, Wiener index, modified Wiener index, hyper-Wiener index, Harary index, generalized Harary index, and multiplicative Wiener index for hierarchical hypercube networks. Our results can help to understand topology of hierarchical hypercube networks and are useful to enhance the ability of these networks. Our result

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40

NURKAHLI, SEMIHA BASDAS, and SERIFE BUYUKKOSE. "WIENER INDEX OF INTERVAL WEIGHTED GRAPHS." Journal of Science and Arts 21, no. 1 (2021): 21–28. http://dx.doi.org/10.46939/j.sci.arts-21.1-a03.

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The Wiener index is classic and well-known topological index for the characterization of molecular graphs. In this paper, we study interval weighted graphs such as graphs where the edge weights are interval or interval matrices. In this study firstly we define interval weighted graph. Later we define Wiener index of interval weighted graph. From this definition, we give algorithms for finding Wiener index of interval weighted directed and undirected graph.

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41

Zhang, Bing, and Bo Zhou. "On Modified and Reverse Wiener Indices of Trees." Zeitschrift für Naturforschung A 61, no. 10-11 (2006): 536–40. http://dx.doi.org/10.1515/zna-2006-10-1104.

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The Wiener index is a well-known measure of graph or network structures with similarly useful variants of modified and reverse Wiener indices. The Wiener index of a tree T obeys the relation W(T)=nT,1(e)·nT,2(e) where nT,1(e) and nT,2(e) are the number of vertices of T lying on the two sides of the edge e, and where the summation goes over all edges of T. The λ -modified Wiener index is defined as mWλ (T) =[nT,1(e)·nT,2(e)]λ . For each λ > 0 and each integer d with 3 ≤ d ≤ n− 2, we determine the trees with minimal λ -modified Wiener indices in the class of trees with n vertices and diameter

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42

Masre, Mesfin, Samuel Asefa Fufa, and Tomáš Vetrík. "Distance-based indices of complete m-ary trees." Discrete Mathematics, Algorithms and Applications 12, no. 04 (2020): 2050041. http://dx.doi.org/10.1142/s179383092050041x.

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Binary and [Formula: see text]-ary trees have extensive applications, particularly in computer science and chemistry. We present exact values of all important distance-based indices for complete [Formula: see text]-ary trees. We solve recurrence relations to obtain the value of the most well-known index called the Wiener index. New methods are used to express the other indices (the degree distance, the eccentric distance sum, the Gutman index, the edge-Wiener index, the hyper-Wiener index and the edge-hyper-Wiener index) as well. Values of distance-based indices for complete binary trees are c

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43

Satriawan, Didit, Qurratul Aini, Abdurahim, Fariz Maulana, and I. Gede Adhitya Wisnu Wardhana. "Molecular Topology Index of a Zero Divisor Graph on a Ring of Integers Modulo Prime Power Order." Contemporary Mathematics and Applications (ConMathA) 6, no. 2 (2024): 72–82. http://dx.doi.org/10.20473/conmatha.v6i2.54737.

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In chemistry, graph theory has been widely utilized to address molecular problems, with numerous applications in graph theory and ring theory within this field. One of these applications involves topological indices that represent chemical structures with numerical values. Various types of topological indices exist, including the Wiener index, the first Zagreb index, and the hyper-Wiener index. In the context of this research, the values of the Wiener index, the first Zagreb index, and the hyper-Wiener index for zero-divisor graphs on the ring of integers modulo a prime power order will be exp

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Essalih, Mohamed, Mohamed El Marraki, and Abd Errahmane Atmani. "The Wiener index, the hyper-Wiener index and the degree distance index of the corona Cm.Cn." Applied Mathematical Sciences 8 (2014): 4217–26. http://dx.doi.org/10.12988/ams.2014.43202.

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Klavžar, Sandi, and M. J. Nadjafi-Arani. "Wiener index versus Szeged index in networks." Discrete Applied Mathematics 161, no. 7-8 (2013): 1150–53. http://dx.doi.org/10.1016/j.dam.2012.12.007.

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Feng, Lihua, Xiaomin Zhu, and Weijun Liu. "Wiener index, Harary index and graph properties." Discrete Applied Mathematics 223 (May 2017): 72–83. http://dx.doi.org/10.1016/j.dam.2017.01.028.

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Tratnik, Niko. "A method for computing the edge-hyper-Wiener index of partial cubes and an algorithm for benzenoid systems." Applicable Analysis and Discrete Mathematics 12, no. 1 (2018): 126–42. http://dx.doi.org/10.2298/aadm1801126t.

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The edge-hyper-Wiener index of a connected graph G is defined as WWe(G)=1/2 ?{e,f}?E(G) d(e,f) + 1/2 ? {e,f}?E(G)d(e,f)2. We develop a method for computing the edge-hyper-Wiener index of partial cubes, which constitute a large class of graphs with a lot of applications. It is also shown how the method can be applied to trees. Furthermore, an algorithm for computing the edge-hyper-Wiener index of benzenoid systems is obtained. Moreover, the algorithm is used to recalculate already known closed formulas for the edge-Wiener index and the edge-hyper-Wiener index of linear polyacenes. Finally, the

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Alex, L., and Indulal G. "ON THE WIENER INDEX OF F_H SUMS OF GRAPHS." Journal of Computer Science and Applied Mathematics 3, no. 2 (2021): 37–57. http://dx.doi.org/10.37418/jcsam.3.2.1.

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Wiener index is the first among the long list of topological indices which was used to correlate structural and chemical properties of molecular graphs. In \cite{Eli} M. Eliasi, B. Taeri defined four new sums of graphs based on the subdivision of edges with regard to the cartesian product and computed their Wiener index. In this paper, we define a new class of sums called $F_H$ sums and compute the Wiener index of the resulting graph in terms of the Wiener indices of the component graphs so that the results in \cite{Eli} becomes a particular case of the Wiener index of $F_H$ sums for $H = K_1$

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Xia, Hong, Liu Qi, and Feng Wei. "The Wiener Index of Two Cubic Graphs." Journal of Physics: Conference Series 3004, no. 1 (2025): 012025. https://doi.org/10.1088/1742-6596/3004/1/012025.

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Abstract The Wiener index is not only one of the most commonly used molecular topological index, but also an important chemical index in chemical graph theory. The Wiener index W (G) of a graph G is ∑ { u , v } ⊆ V ( G ) d G ( u , v ) . In this paper, we determine the exact value of Wiener index of P n 3 and C n 3 by mathematical induction and classification discussion, thus enriching the topological index theory.

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Hossein-Zadeh, S., A. Hamzeh, and A. R. Ashrafi. "Wiener-type invariants of some graph operations." Filomat 23, no. 3 (2009): 103–13. http://dx.doi.org/10.2298/fil0903103h.

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Let d(G, k) be the number of pairs of vertices of a graph G that are at distance k, ? a real number, and W?(G) =?k?1 d(G, k)k?. W?(G) is called the Wiener-type invariant of G associated to real number ?. In this paper, the Wiener-type invariants of some graph operations are computed. As immediate consequences, the formulae for reciprocal Wiener index, Harary index, hyper- Wiener index and Tratch-Stankevich-Zefirov index are calculated. Some upper and lower bounds are also presented.

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