The maximum hyper-Wiener index of cacti

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• 作者：Dong-fang Wang (1)
  Shang-wang Tan (1)
  
  1. Department of Mathematics ; China University of Petroleum ; Qingdao聽 ; 266580 ; People鈥檚 Republic of China
  
• 关键词：Cacti ; Wiener index ; Hyper ; Wiener index ; Distance ; Pendant path ; 05C90 ; 05C12 ; 05C35
• 刊名：Journal of Applied Mathematics and Computing
• 出版年：2015
• 出版时间：February 2015
• 年：2015
• 卷：47
• 期：1-2
• 页码：91-102
• 全文大小：306 KB
• 参考文献：1. Wiener, H.: Structural determination of paraffin boiling points. J. Am. Chem. Soc. 69, 17鈥?0 (1947) CrossRef
  2. Hosoya, H.: Topological index, a newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons. Bull. Chem. Soc. Jpn. 44, 2332鈥?339 (1971) CrossRef
  3. Dobrynin, A.A., Entringer, R., Gutman, I.: Wiener index of trees: theory and applications. Acta Appl. Math. 66, 211鈥?49 (2001) CrossRef
  4. Randi膰, M.: Novel molecular descriptor for structure鈥損roperty studies. Chem. Phys. Lett. 211, 478鈥?83 (1993) CrossRef
  5. Tan, S.W., Wang, Q.L.: The Wiener index of cacti given matching number. Ars Comb. (in press)
  6. Liu, H.Q., Lu, M.: A unified approach to extremal cacti for different indices. MATCH Commun. Math. Comput. Chem. 58, 183鈥?94 (2007)
  7. Feng, L.H., Liu, W.J., Xu, K.X.: The hyper-Wiener index of bicyclic graphs. Utilitas Math. 84, 97鈥?04 (2011)
  8. Feng, L.H., Yu, G.H.: On the hyper-Wiener index of cacti, Utilitas Math., (in press)
  9. Xing, R.D., Zhou, B., Qi, X.L.: Hyper-Wiener Index of Unicyclic Graphs. MATCH Commun. Math. Comput. Chem. 66, 315鈥?28 (2011)
  10. Gutman, I., Linert, W., Lukovits, I., Dobrynin, A.A.: Trees with external hyper-Wiener index: mathematical basis and chemical applications. J. Chem. Inf. Comput. Sci. 37, 349鈥?54 (1997) CrossRef
  11. Gutman, I., Furtula, B.: Hyper-Wiener index versus Wiener index vs. Two highly correlated structure-descriptors. Monatsh. Chem. 134, 975鈥?81 (2003)
  12. Gutman, I.: Relation between hyper-Wiener and Wiener index. Chem. Phys. Lett. 364, 352鈥?56 (2002) CrossRef
  13. Liu, M.H., Liu, B.L.: Trees with the seven smallest and fifteen greastest hyper-Wiener indices. MATCH Commun. Math. Comput. Chem. 63, 151鈥?70 (2010)
  14. Xu, K.X., Trinajsti膰, N.: Hyper-Wiener and Harary indices of graphs with cut edges. Utilitas Math. 84, 153鈥?63 (2011)
  15. Todeschini, R., Consonni, V.: Molecular Descriptors for Chemoinformatics. Wiley, Weinheim (2009) CrossRef
  16. Wang, H.Z., Kang, L.Y.: More on the Harary index of cacti. J. Appl. Math. Comput. 43, 369鈥?86 (2013) CrossRef
  17. Das, K.: Some algorithms on cactus graphs. Annal. Pure Appl. Math. 2(2), 114鈥?28 (2012)
  18. Zmazek, B., 沤erovnik, J.: Estimating the traffic on weighted cactus networks in linear time. In: Ninth International Conference on Information Visualization, pp. 536鈥?41, London (2005)
  19. Xu, K., Wang, J., Liu, H.: The Harary index of ordinary and generalized quasi-tree graphs. J. Appl. Math. Comput. doi:10.1007/s12190-013-0727-4 (2013)
  20. Dobrynin, A.A., Gutman, I., Klav啪ar, S., 沤igert, P.: Wiener index of hexagonal systems. Acta Appl. Math. 72(3), 247鈥?94 (2002) CrossRef
  21. Tang, Z.K., Deng, H.Y.: The \((n,n)\) -graphs with the first three extremal Wiener indices. J. Math. Chem. 43(1), 60鈥?4 (2008) CrossRef
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Computational Mathematics and Numerical Analysis
  Applied Mathematics and Computational Methods of Engineering
  Theory of Computation
  Mathematics of Computing
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1865-2085

文摘

The Wiener index of a connected graph \(G\) is the sum of distances between all unordered pairs of vertices in the graph. The hyper-Wiener index is defined as \(WW(G)= \frac{1}{2}\sum \nolimits _{\{u,v\} \subseteq V(G)}( d(u,v)+d^2 (u,v))\) , where \(d(u,v)\) is the number of edges on a shortest path connecting vertices \(u\) and \(v\) . A cactus graph is a connected graph in which each block is either an edge or a cycle. In the paper, we characterize the extremal cacti having the largest Wiener and hyper-Wiener indexes among all cacti with \(n\) vertices and \(r\) cycles.