Term of Award
Summer 2018
Degree Name
Master of Science in Mathematics (M.S.)
Document Type and Release Option
Thesis (open access)
Copyright Statement / License for Reuse
This work is licensed under a Creative Commons Attribution 4.0 License.
Department
Department of Mathematical Sciences
Committee Chair
Hua Wang
Committee Member 1
Daniel Grey
Committee Member 2
Jie Zhang
Abstract
In this thesis, we consider the properties of sparse trees and summarized a certain class of trees under some constraint (including with a given degree sequence, with given number of leaves, with given maximum degree, etc.) which have maximum Wiener index and the minimum number of subtrees at the same time. Wiener index is one of the most important topological indices in chemical graph theory. Steiner k�� Wiener index can be regarded as the generalization of Wiener index, when k = 2, Steiner Wiener index is the same as Wiener index. Steiner k�� Wiener index of a tree T is the summation of all sizes of subtrees which contain any k��subset of vertex set V (T). In sparse trees with a given degree sequence, by using a computer, we give some computational results with regarding to Steiner Wiener index, which may shed light on the extremal tree with maximum Steiner Wiener index.
Recommended Citation
[1] E. Andriantiana, S.Wagner, H.Wang, Greedy trees, subtrees and antichains, Electron. J. Combin., 20 (2013), P28. [2] J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, Macmillan (London) (1976). [3] P. Dankelmann, O. R. Oellermann, H. C. Swart, The average Steiner distance of a Graph, Journal of Graph Theory, 22 (1) (1996), 15–22. [4] P. Dankelmann, H. C. Swart, O. R. Oellermann, On the average Steiner distance of graphs with prescribed properties, Discrete Appl. Math., 79 (1997), 91–103. [5] A. Dobrynin, R. Entringer, I. Gutman, Wiener index of trees: Theory and applications, Acta Appl. Math., 66 (2001), 211–249. [6] R. C. Entringer, Bounds for the average distance-inverse degree product in trees, in: Y. Alavi, D. R. Lick, A. J. Schwenk (Eds.), Combinatorics, Graph Theory, and Algorithms, New Issues Press, Kalamazoo, 1999, 335–352. [7] M. Fischermann, A. Hoffmann, D. Rautenbach, L. Sz´ekely, L. Volkmann, Wiener index versus maximum degree in trees, Discrete Appl. Math., 122 (2002), 127–137. [8] I. Peterin, P. Z. Pleterˇsek, Wiener index of strong product of graphs, Opuscula Math., 38 (1) (2018), 81–94. [9] X. Li, Y. Mao, I. Gutman, The Steiner Wiener index of a graph, Discuss. Math. Graph Theory, 36 (2016), 455-465. [10] X. Li, Y. Mao, I. Gutman, Inverse problem on the Steiner Wiener index, Discuss. Math. Graph Theory, 38 (2018), 83–95. [11] L. Lu, Q. Huang, J. Hou, X. Chen, A sharp lower bound on Steiner Wiener index for trees with given diameter, Discrete Mathematics, 341 (2018), 723–731. [12] N. Schmuck, S. Wagner, H. Wang, Greedy trees, caterpillars, and Wiener-type graph invariants, MATCH Commun. Math. Comput. Chem., 68 (2012), 273–292. [13] R. Shi, The average distance of trees, Systems Sci. Math. Sci., 6 (1) (1993), 18–24. [14] L. A. Sz´ekely, H. Wang, Binary trees with the largest number of subtrees, Discrete Appl. Math., 155 (2007), 374–385. [15] L. A. Sz´ekely, H. Wang, On subtrees of trees, Adv. in Appl. Math., 34 (2005), 138– 155. [16] L. A. Sz´ekely, H. Wang, Extremal values of ratios: distance problems vs. subtree problems in trees, The Electronic Journal of Combinatorics, 20(1) (2013), P67. [17] L. A. Sz´ekely, H. Wang, Extremal values of ratios: distance problems vs. subtree problems in trees II, Discrete Math., 322 (2014), 36–47. [18] S. Wagner, On the Wiener index of random trees, Discrete Math., 312 (9) (2012), 1502–1511. [19] H. Wang, The extremal values of the Wiener index of a tree with given degree sequence, Discrete Appl. Math., 156 (2008), 2647–2654. [20] S.Wang, X. Guo, Trees with extremalWiener indices, MATCH Commun. Math. Comput. Chem., 60 (2008), 609–622. [21] H.Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc., 69 (1947), 17–20. [22] H. Wiener, Correlation of heats of isomerization, and differences in heats of vaporization of isomers, among the paraffin hydrocarbons, J. Am. Chem. Soc., 69 (1947), 2636–2638. [23] W. Yan, Y-N. Yeh, Enumeration of subtrees of trees, Theoretical Computer Science, 369 (2006), 256–268. [24] J. Zhang, H. Wang, X.-D. Zhang, Trees, degree sequences, and the Steiner Wiener index, Preprint. [25] J. Zhang, H. Wang, X.-D. Zhang, Revisiting the Wiener index and the number of subtrees, Preprint. [26] X.-D. Zhang, Q.-Y. Xiang, L.-Q. Xu, R.-Y. Pan, The Wiener index of trees with given degree sequences, MATCH Commun. Math. Comput. Chem., 60 (2008), 623–644. [27] X.-M. Zhang, X.-D. Zhang, The minimal number of subtrees with a given degree sequence, Graphs and Combinatorics, 31 (2015), 309–318. [28] X.-M. Zhang, X.-D. Zhang, D. Gray, H. Wang, The number of subtrees of trees with given degree sequence, J. Graph Theory, 73 (2013), 280–295.
Research Data and Supplementary Material
No
