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Problems related to graph indices in trees

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Recent Trends in Combinatorics

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 159))

Abstract

In this chapter we explore recent development on various problems related to graph indices in trees. We focus on indices based on distances between vertices, vertex degrees, or on counting vertex or edge subsets of different kinds. Some of the indices arise naturally in applications, e.g., in chemistry, statistical physics, bioinformatics, and other fields, and connections are also made to other branches of graph theory, such as spectral graph theory. We will be particularly interested in the extremal values (maxima and minima) for different families of trees and the corresponding extremal trees. Moreover, we review results for random trees, consider localized versions of different graph indices and the associated notions of centrality, and finally discuss inverse problems, where one wants to find trees for which a specific graph index has a prescribed value.

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References

  1. T. Ali Khan, R. Neininger, Tail bounds for the Wiener index of random trees, in Discrete Math. Theor. Comput. Sci. Proc., Proceedings of the 2007 Conference on the Analysis of Algorithms (2007), pp. 279–289

    Google Scholar 

  2. N. Alon, S. Haber, M. Krivelevich, The number of F-matchings in almost every tree is a zero residue. Electron. J. Comb. 18(1), paper no. #P30 (2011)

    Google Scholar 

  3. M. Aouchiche, P. Hansen, M. Zheng, Variable neighborhood search for extremal graphs. 19. Further conjectures and results about the Randić index. MATCH Commun. Math. Comput. Chem. 58, 83–102 (2007)

    Google Scholar 

  4. E.O.D. Andriantiana, Energy, Hosoya index and Merrifield-Simmons index of trees with prescribed degree sequence. Discrete Appl. Math. 161(6), 724–741 (2013)

    MathSciNet  MATH  Google Scholar 

  5. E.O.D. Andriantiana, S. Wagner, Spectral moments of trees with given degree sequence. Linear Algebra Appl. 439(12), 3980–4002 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. E.O.D. Andriantiana, S. Wagner, Trees with minimum number of infima closed sets. In preparation (2015)

    Google Scholar 

  7. E.O.D. Andriantiana, S. Wagner, H. Wang, Greedy trees, subtrees and antichains. Electron. J. Comb. 20(3), paper no. #P28 (2013)

    Google Scholar 

  8. A.T. Balaban, Chemical Applications of Graph Theory (Academic Press, New York, 1976)

    Google Scholar 

  9. C.A. Barefoot, R.C. Entringer, L.A. Székely, Extremal values for ratios of distances in trees. Discrete Appl. Math. 80, 37–56 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  10. R.J. Baxter, I.G. Enting, S.K. Tsang, Hard-square lattice gas. J. Stat. Phys. 22(4), 465–489 (1980)

    Article  MathSciNet  Google Scholar 

  11. S. Bereg, H. Wang, Wiener indices of balanced binary trees. Discrete Appl. Math. 155, 457–467 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. N.L. Biggs, E.K. Lloyd, R.J. Wilson, Graph Theory 1736–1936 (Clarendon Press, Oxford, 1976; Oxford University Press, Oxford, 1986)

    Google Scholar 

  13. T. Bıyıkoğlu, J. Leydold, Graphs with given degree sequence and maximal spectral radius. Electron. J. Comb. 15(1), paper no. #P119 (2008)

    Google Scholar 

  14. D. Bokal, É. Czabarka, L.A. Székely, I. Vrťo, General lower bounds for the minor crossing numbers of graphs. Discrete Comput. Geom. 44, 463–483 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. B. Bollobás, P. Erdős, Graphs of extremal weights. Ars Comb. 50, 225–233 (1998)

    MathSciNet  MATH  Google Scholar 

  16. B. Bollobás, P. Erdős, A. Sarkar, Extremal graphs for weights. Discrete Math. 200, 5–19 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  17. B. Bollobás, M. Tyomkyn, Walks and paths in trees. J. Graph Theory 70(1), 54–66 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. D. Bród, Z. Skupień, Trees with extremal numbers of dominating sets. Australas. J. Comb. 35, 273–290 (2006)

    MathSciNet  MATH  Google Scholar 

  19. D. Bród, Z. Skupień, Recurrence among trees with most numerous efficient dominating sets. Discrete Math. Theor. Comput. Sci. 10(1), 43–55 (2008)

    MathSciNet  MATH  Google Scholar 

  20. A.E. Brouwer, The number of dominating sets of a finite graph is odd. Preprint (2009). http://www.win.tue.nl/~aeb/preprints/domin2.pdf

  21. V. Bruyère, H. Mélot, Fibonacci index and stability number of graphs: a polyhedral study. J. Comb. Optim. 18(3), 207–228 (2009)

    Google Scholar 

  22. V. Bruyère, G. Joret, H. Mélot, Trees with given stability number and minimum number of stable sets. Graphs Comb. 28(2), 167–187 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. V. Campos, V. Chvátal, L. Devroye, P. Taslakian, Transversals in trees. J. Graph Theory 73(1), 32–43 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. G. Caporossi, P. Hansen, Variable neighborhood search for extremal graphs 1: the AutographiX system. Discrete Math. 212, 29–44 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  25. G. Caporossi, I. Gutman, P. Hansen, L. Pavlović, Graphs with maximum connectivity index. Comput. Biol. Chem. 27, 85–90 (2000)

    Article  Google Scholar 

  26. A. Cayley, On the mathematical theory of isomers. Philos. Mag. 47, 444–446 (1874)

    Google Scholar 

  27. A. Cayley, A theorem on trees. Quart. J. Math. 23, 376–378 (1889)

    MATH  Google Scholar 

  28. E. Çela, N. Schmuck, S. Wimer, G.J. Woeginger, The Wiener maximum quadratic assignment problem. Discrete Optim. 8, 411–416 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. F.R.K. Chung, Spectral graph theory, in CBMS Regional Conference Series in Mathematics, vol. 92 (American Mathematical Society, Providence, 1997)

    Google Scholar 

  30. L. Clark, I. Gutman, The exponent in the general Randić index. J. Math. Chem. 43 (1), 32–44 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. P. Csikvári, On a poset of trees. Combinatorica 30(2), 125–137 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. P. Csikvári, On a poset of trees II. J. Graph Theory 74(1), 81–103 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  33. D.M. Cvetković, I. Gutman, (eds.), Selected topics on applications of graph spectra. Zb. Rad. (Beogr.) 14(22), 155–174 (2011)

    Google Scholar 

  34. D.M. Cvetković, M. Doob, H. Sachs, Spectra of Graphs. Theory and Applications, 3rd edn. (Johann Ambrosius Barth, Heidelberg, 1995)

    Google Scholar 

  35. D.M. Cvetković, P. Rowlinson, S. Simić, An Introduction to the Theory of Graph Spectra. London Mathematical Society Student Texts, vol. 75 (Cambridge University Press, Cambridge, 2010)

    Google Scholar 

  36. É. Czabarka, L.A. Székely, S. Wagner, The inverse problem for certain tree parameters. Discrete Appl. Math. 157(15), 3314–3319 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  37. É. Czabarka, L.A. Székely, S. Wagner, On the number of nonisomorphic subtrees of a tree. (2016, submitted, arXiv:1601.00944)

    Google Scholar 

  38. É. Czabarka, A. Dutle, P.L. Erdős, I. Miklós, On realizations of a Joint Degree Matrix. Discrete Appl. Math. 181, 283–288 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  39. C. Delorme, O. Favaron, D. Rautenbach, Closed formulas for the numbers of small independent sets and matchings and an extremal problem for trees. Discrete Appl. Math. 130, 503–512 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  40. H. Deng, Q. Guo, On the minimal Merrifield-Simmons index of trees of order n with at least \([n/2] + 1\) pendent vertices. MATCH Commun. Math. Comput. Chem. 60(2), 601–608 (2001)

    MathSciNet  MATH  Google Scholar 

  41. A.A. Dobrynin, I. Gutman, The average Wiener index of trees and chemical trees. J. Chem. Inf. Comput. Sci. 39, 679–683 (1999)

    Article  Google Scholar 

  42. A.A. Dobrynin, R. Entringer, I. Gutman, Wiener index of trees. Theory and applications. Acta Appl. Math. 66, 211–249 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  43. M. Drmota, Random Trees. An Interplay Between Combinatorics and Probability (Springer, Wien/New York, 2009)

    Google Scholar 

  44. R.C. Entringer, D.E. Jackson, D.A. Snyder, Distance in graphs. Czech. Math. J. 26, 283–296 (1976)

    MathSciNet  MATH  Google Scholar 

  45. R.C. Entringer, A. Meir, J.W. Moon, L.A. Székely, On the Wiener index of trees from certain families. Australas. J. Comb. 10, 211–224 (1994)

    MathSciNet  MATH  Google Scholar 

  46. S. Fajtlowicz, On conjectures of Graffiti. Discrete Math. 72, 113–118 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  47. E.J. Farrell, An introduction to matching polynomials. J. Comb. Theory Ser. B 27(1), 75–86 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  48. Q. Feng, H.M. Mahmoud, A. Panholzer, Limit laws for the Randić index of random binary tree models. Ann. Inst. Stat. Math. 60(2), 319–343 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  49. Q. Feng, Z. Hu, On the Zagreb index of random recursive trees. J. Appl. Probab. 48(4), 1189–1196 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  50. Q. Feng, Z. Hu, Phase changes in the topological indices of scale-free trees. J. Appl. Probab. 50(2), 516–532 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  51. J.A. Fill, S. Janson, Precise logarithmic asymptotics of the right tails of some limit random variables for random trees. Ann. Comb. 12, 403–416 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  52. M. Fischermann, A. Hoffmann, D. Rautenbach, L. Szekely, L. Volkmann, Wiener index versus maximum degree in trees. Discrete Appl. Math. 122, 127–137 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  53. M. Fischermann, L. Volkmann, D. Rautenbach, A note on the number of matchings and independent sets in trees. Discrete Appl. Math. 145(3), 483–489 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  54. M. Fuchs, C.-K. Lee, The Wiener index of random digital trees. SIAM J. Discrete Math. 29(1), 586–614 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  55. D. Goldman, S. Istrail, G. Lancia, A. Piccolboni, B. Walenz, Algorithmic strategies in combinatorial chemistry, in SODA ’00 Proceedings of the eleventh annual ACM-SIAM Symposium on Discrete Algorithms (2000), pp. 275–284

    Google Scholar 

  56. J. Górska, Z. Skupień, Trees with maximum number of maximal matchings. Discrete Math. 307(11–12), 1367–1377 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  57. I. Gutman, A property of the Wiener number and its modifications. Indian J. Chem. 36A, 128–132 (1997)

    Google Scholar 

  58. I. Gutman, Degree-based topological indices. Croat Chem. Acta 86(4), 351–361 (2013)

    Article  Google Scholar 

  59. I. Gutman, W. Linert, I. Lukovits, A.A. Dobrynin, Trees with extremal hyper-Wiener index: Mathematical basis and chemical applications. J. Chem. Inf. Comput. Sci. 37, 349–354 (1997)

    Article  Google Scholar 

  60. I. Gutman, O. Miljković, G. Caporossi, P. Hansen, Alkanes with small and large Randić connectivity indices. Chem. Phys. Lett. 306(5), 366–372 (1999)

    Article  Google Scholar 

  61. I. Gutman, B. Furtula, M. Petrović, Terminal Wiener index. J. Math. Chem. 46, 522–531 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  62. I. Gutman, H. Deng, S. Radenković, The Estrada index: an updated survey. Zb. Rad. (Beogr.) 14(22), 155–174 (2011)

    Google Scholar 

  63. J. Haslegrave, Extremal results on average subtree density of series-reduced trees. J. Comb. Theory Ser. B 107, 26–41 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  64. O.J. Heilmann, E.H. Lieb, Theory of monomer-dimer systems. Commun. Math. Phys. 25(3), 190–232 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  65. C. Heuberger, S. Wagner, The number of maximum matchings in a tree. Discrete Math. 311(21), 2512–2542 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  66. H. Hosoya, Topological index. A newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons. Bull. Chem. Soc. Jpn. 44(9), 2332–2339 (1971)

    Google Scholar 

  67. Y. Hou, On acyclic systems with minimal Hosoya index. Discrete Appl. Math. 119(3), 251–257 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  68. Y. Hu, X. Li, Y. Yuan, Trees with minimum general Randic index. MATCH Commun. Math. Comput. Chem. 52, 119–128 (2004)

    MathSciNet  MATH  Google Scholar 

  69. Y. Hu, X. Li, Y. Yuan, Solutions to two unsolved questions on the best upper bound for the Randic index R −1 of trees. MATCH Commun. Math. Comput. Chem. 54, 441–454 (2005)

    MathSciNet  MATH  Google Scholar 

  70. A. Ilić, On the extremal properties of the average eccentricity. Comput. Math. Appl. 64(9), 2877–2885 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  71. O. Ivanciuc, T.S. Balaban, A.T. Balaban, Reciprocal distance matrix, related local vertex invariants and topological indices. J. Math. Chem. 12, 309–318 (1993)

    Article  MathSciNet  Google Scholar 

  72. R.E. Jamison, On the average number of nodes in a subtree of a tree. J. Comb. Theory Ser. B 35(3), 207–223 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  73. R.E. Jamison, Monotonicity of the mean order of subtrees. J. Comb. Theory Ser. B 37(1), 70–78 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  74. S. Janson, The Wiener index of simply generated random trees. Random Struct. Algoritm. 22(4), 337–358 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  75. S. Janson, P. Chassaing, The center of mass of the ISE and the Wiener index of trees. Electron. Commun. Probab. 9, 178–187 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  76. F. Jelen, E. Triesch, Superdominance order and distance of trees with bounded maximum degree. Discrete Appl. Math. 125 (2–3), 225–233 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  77. C. Jordan, Sur les assemblages de lignes. J. Reine Angew. Math. 70, 185–190 (1869)

    Article  MathSciNet  Google Scholar 

  78. L.B. Kier, L.H. Hall, Molecular Connectivity in Chemistry and Drug Research (Academic Press, New York, 1976)

    Google Scholar 

  79. L.B. Kier, L.H. Hall, Molecular Connectivity in Structure-Activity Analysis (Wiley, New York, 1986)

    Google Scholar 

  80. G. Kirchoff, Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme gefürt wird. Ann. Phys. Chem. 72, 497–508 (1847)

    Article  Google Scholar 

  81. R. Kirk, H. Wang, Largest number of subtrees of trees with a given maximum degree. SIAM J. Discrete Math. 22(3), 985–995 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  82. P. Kirschenhofer, H. Prodinger, R.F. Tichy, Fibonacci numbers of graphs. III. Planted plane trees, in Fibonacci Numbers and Their Applications (Patras, 1984) (Reidel, Dordrecht, 1986). Math. Appl. 28, 105–120

    Google Scholar 

  83. M. Klazar, Twelve countings with rooted plane trees. Eur. J. Comb. 18(2), 195–210 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  84. D.J. Klein, I. Lukovits, I. Gutman, On the definition of the hyper-Wiener index for cycle-containing structures. J. Chem. Inf. Comput. Sci. 35, 50–52 (1995)

    Article  Google Scholar 

  85. B. Knudsen, Optimal Multiple Parsimony Alignment with Affine Gap Cost Using a Phylogenetic Tree. Lecture Notes in Bioinformatics, vol. 2812 (Springer, New York, 2003), pp. 433–446

    Google Scholar 

  86. M. Krzywkowski, Trees having many minimal dominating sets. Inform. Process. Lett. 113(8), 276–279 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  87. M. Krzywkowski, Minimal 2-dominating sets in trees. RAIRO Theor. Inform. Appl. 47(3), 235–240 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  88. M. Krzywkowski, An algorithm for listing all minimal double dominating sets of a tree. Fund. Inform. 130(4), 415–421 (2014)

    MathSciNet  MATH  Google Scholar 

  89. M. Krzywkowski, S. Wagner, Trees having few total dominating sets. In preparation (2015)

    Google Scholar 

  90. H.-F. Law, On the number of independent sets in a tree. Electron. J. Comb. 17(1), paper no. #P18 (2010)

    Google Scholar 

  91. M. Lepović, I. Gutman, A collective property of trees and chemical trees. J. Chem. Inf. Comput. Sci. 38, 823–826 (1998)

    Article  Google Scholar 

  92. V.E. Levit, E. Mandrescu, The independence polynomial of a graph—a survey, in Proceedings of the 1st International Conference on Algebraic Informatics (Aristotle University of Thessaloniki, Thessaloniki, 2005), pp. 233–254

    Google Scholar 

  93. S. Li, S. Wang, Further analysis on the total number of subtrees of trees. Electron. J. Comb. 19(4), paper no. #P48 (2012)

    Google Scholar 

  94. X. Li, Y. Shi, A survey on the Randić index. MATCH Commun. Math. Comput. Chem. 59(1), 127–156 (2008)

    MathSciNet  MATH  Google Scholar 

  95. X. Li, I. Gutman, Mathematical Aspects of Randić-type Molecular Structure Descriptors. Mathematical Chemistry Monographs, vol. 1 (University of Kragujevac, Kragujevac, 2006)

    Google Scholar 

  96. X. Li, Z. Li, L. Wang, The inverse problems for some topological indices in combinatorial chemistry. J. Comput. Biol. 10 (1), 47–55 (2003)

    Article  Google Scholar 

  97. X. Li, Y. Shi, I. Gutman, Graph Energy (Springer, New York, 2012)

    Book  MATH  Google Scholar 

  98. V. Linek, Bipartite graphs can have any number of independent sets. Discrete Math. 76 (2), 131–136 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  99. L. Lovász, Combinatorial Problems and Exercises (North-Holland, Amsterdam, 1979)

    MATH  Google Scholar 

  100. L. Lu, Y. Yang, A theorem on Randić index and the diameter of a graph. Discrete Math. 311(14), 1333–1343 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  101. A. Meir, J.W. Moon, On subtrees of certain families of rooted trees. Ars Comb. 16B, 305–318 (1983)

    MathSciNet  MATH  Google Scholar 

  102. A. Meir, J.W. Moon, On maximal independent sets of nodes in trees. J. Graph Theory 12(2), 265–283 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  103. R.E. Merrifield, H.E. Simmons, Topological Methods in Chemistry (Wiley, New York, 1989)

    Google Scholar 

  104. G.O. Munsonius, On the asymptotic internal path length and the asymptotic Wiener index of random split trees. Electron. J. Probab. 16 paper no. 35, 1020–1047 (2011)

    Google Scholar 

  105. G.O. Munsonius, On tail bounds for random recursive trees. J. Appl. Probab. 49, 566–581 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  106. R. Neininger, The Wiener index of random trees. Comb. Probab. Comput. 11(6), 587–597 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  107. V. Nikiforov, Walks and the spectral radius of graphs. Linear Algebra Appl. 418(1), 257–268 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  108. L. Pavlovic, M. Stojanovic, X. Li, More on “Solutions to two unsolved questions on the best upper bound for the Randić Index R −1 of trees”. MATCH Commun. Math. Comput. Chem. 58(1), 167–182 (2007)

    MathSciNet  MATH  Google Scholar 

  109. D. Plavsić, S. Nikolić, N. Trinajstić, Z. Mihalić, On the Harary index for the characterization of chemical graphs. J. Math. Chem. 12, 235–250 (1993)

    Article  MathSciNet  Google Scholar 

  110. H. Prodinger, R.F. Tichy, Fibonacci numbers of graphs. Fibonacci Quart. 20(1), 16–21 (1982)

    MathSciNet  MATH  Google Scholar 

  111. M. Randić, Characterization of molecular branching. J. Am. Chem. Soc. 97 (23), 6609–6615 (1975)

    Article  Google Scholar 

  112. M. Randić, Novel molecular descriptor for structure-property studies. Chem. Phys. Lett. 211, 478–483 (1993)

    Article  Google Scholar 

  113. B.E. Sagan, A note on independent sets in trees. SIAM J. Discrete Math. 1(1), 105–108 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  114. B.E. Sagan, V.R. Vatter, Maximal and maximum independent sets in graphs with at most r cycles. J. Graph Theory 53(4), 283–314 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  115. N.S. Schmuck, S. Wagner, H. Wang, Greedy trees, caterpillars, and Wiener-type graph invariants. MATCH Commun. Math. Comput. Chem. 68, 273–292 (2012)

    MathSciNet  MATH  Google Scholar 

  116. E. Schroeder, Vier combinatorische Probleme. Z. f. Math. Phys. 15, 361–376 (1870)

    Google Scholar 

  117. R. Shi, The average distance of trees. Syst. Sci. Math. Sci. 6(1), 18–24 (1993)

    MathSciNet  MATH  Google Scholar 

  118. A.V. Sills, H. Wang, On the maximal Wiener index and related questions. Discrete Appl. Math. 160, 1615–1623 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  119. A.V. Sills, H. Wang, The minimal number of subtrees of a tree. Graphs Comb. 31(1), 255–264 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  120. Z. Skupień, Majorization and the minimum number of dominating sets. Discrete Appl. Math. 165, 295–302 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  121. H. Smith, L.A. Székely, H. Wang, Eccentricity in trees (2014, submitted, arXiv:1408.5865)

    Google Scholar 

  122. H. Smith, L.A. Székely, H. Wang, S. Yuan, On different “middle parts” of a tree. In preparation (2015)

    Google Scholar 

  123. L.A. Székely, H. Wang, Binary trees with the largest number of subtrees with at least one leaf. Congr. Numer. 177, 147–169 (2005),

    MathSciNet  MATH  Google Scholar 

  124. L.A. Székely, H. Wang, On subtrees of trees. Adv. Appl. Math. 34, 138–155 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  125. L.A. Székely, H. Wang, Binary trees with the largest number of subtrees. Discrete Appl. Math. 155(3), 374–385 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  126. L.A. Székely, H. Wang, Extremal values of ratios: distance problems vs. subtree problems in trees. Electron. J. Comb. 20(1), #P67, 1–20 (2013)

    Google Scholar 

  127. L.A. Székely, H. Wang, Extremal values of ratios: distance problems vs. subtree problems in trees II. Discrete Math. 322, 36–47 (2014)

    Google Scholar 

  128. L.A. Székely, H. Wang, T. Wu, The sum of distances between the leaves of a tree and the ‘semi-regular’ property. Discrete Math. 311, 1197–1203 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  129. Y. Tang, B. Zhou, On Average Eccentricity. MATCH Commun. Math. Comput. Chem. 67, 405–423 (2012)

    MathSciNet  MATH  Google Scholar 

  130. H. Täubig, J. Weihmann, S. Kosub, R. Hemmecke, E.W. Mayr, Inequalities for the number of walks in graphs. Algorithmica 66(4), 804–828 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  131. H. Täubig, J. Weihmann, Matrix power inequalities and the number of walks in graphs. Discrete Appl. Math. 176, 122–129 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  132. R. Todeschini, V. Consonni, Handbook of Molecular Descriptors (Wiley-VCH, Weinheim, 2000)

    Book  Google Scholar 

  133. N. Trinajstić, Chemical Graph Theory (CRC Press, Boca Raton, 1992)

    Google Scholar 

  134. A. Vince, H. Wang, The average order of a subtree of a tree. J. Comb. Theory Ser. B 100(2), 161–170 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  135. S. Wagner, A class of trees and its Wiener index. Acta Appl. Math. 91(2), 119–132 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  136. S. Wagner, Correlation of graph-theoretical indices. SIAM J. Discrete Math. 21(1), 33–46 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  137. S. Wagner, On the average Wiener index of degree-restricted trees. Australas. J. Comb. 37, 187–203 (2007)

    MathSciNet  MATH  Google Scholar 

  138. S. Wagner, On the number of matchings of a tree. Eur. J. Comb. 28(4), 1322–1330 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  139. S. Wagner, Extremal trees with respect to Hosoya index and Merrifield-Simmons index. MATCH Commun. Math. Comput. Chem. 57(1), 221–233 (2007)

    MathSciNet  MATH  Google Scholar 

  140. S. Wagner, Almost all trees have an even number of independent sets. Electron. J. Comb. 16(1) paper no. #93 (2009)

    Google Scholar 

  141. S. Wagner, A note on the number of dominating sets of a graph. Util. Math. 92, 25–31 (2013)

    MathSciNet  MATH  Google Scholar 

  142. S. Wagner, Central limit theorems for additive tree parameters with small toll functions. Comb. Probab. Comput. 24(1), 329–353 (2015)

    Article  MathSciNet  Google Scholar 

  143. S. Wagner, I. Gutman, Maxima and minima of the Hosoya index and the Merrifield-Simmons index: a survey of results and techniques. Acta Appl. Math. 112(3), 323–346 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  144. S. Wagner, H. Wang, Indistinguishable trees and graphs. Graphs Comb. 30(6), 1593–1605 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  145. S. Wagner, H. Wang, On the local and global means of subtree orders. J. Graph Theory 85(2), 154–166 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  146. S. Wagner, H. Wang, G. Yu, Molecular graphs and the inverse Wiener index problem, Discrete Appl. Math. 157(7), 1544–1554 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  147. S. Wagner, H. Wang, X.D. Zhang, Distance-based graph invariants of trees and the Harary index. Filomat. 27, 39–48 (2013)

    Article  MathSciNet  Google Scholar 

  148. H. Wang, Extremal trees with given degree sequence for the Randić index. Discrete Math. 308, 3407–3411 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  149. H. Wang, The extremal values of the Wiener index of a tree with given degree sequence. Discrete Appl. Math. 156, 2647–2654 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  150. H. Wang, Functions on adjacent vertex degrees of trees with given degree sequence. Cent. Eur. J. Math. 12, 1656–1663 (2014)

    MathSciNet  MATH  Google Scholar 

  151. H. Wang, The distances between internal vertices and leaves of a tree. Eur. J. Comb. 41, 79–99 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  152. H. Wang, G. Yu, All but 49 numbers are Wiener indices of trees. Acta Appl. Math. 92(1) 15–20 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  153. H. Wiener, Structural determination of paraffin boiling points. J. Am. Chem. Soc. 69, 17–20 (1947)

    Article  Google Scholar 

  154. H. Wiener, Correlation of heats of isomerization, and differences in heats of vaporization of isomers, among the paraffin hydrocarbons. J. Am. Chem. Soc. 69, 2636–2638 (1947)

    Article  Google Scholar 

  155. H.S. Wilf, The number of maximal independent sets in a tree. SIAM J. Algebraic Discrete Methods 7(1), 125–130 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  156. I. Włoch, Trees with extremal numbers of maximal independent sets including the set of leaves. Discrete Math. 308(20), 4768–4772 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  157. I. Włoch, A. Włoch, The number of independent sets intersecting the set of leaves in trees. Ars Comb. 85, 225–231 (2007)

    MathSciNet  MATH  Google Scholar 

  158. K. Xu, Trees with the seven smallest and eight greatest Harary indices. Discrete Appl. Math. 160, 321–331 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  159. K. Xu, M. Liu, K. Das, I. Gutman, B. Furtula, A survey on graphs extremal with respect to distance-based topological indices. MATCH Commun. Math. Comput. Chem. 71, 461–508 (2014)

    MathSciNet  Google Scholar 

  160. W. Yan, L. Ye, On the maximal energy and the Hosoya index of a type of trees with many pendant vertices. MATCH Commun. Math. Comput. Chem. 53(2), 449–459 (2005)

    MathSciNet  MATH  Google Scholar 

  161. P. Yu, An upper bound on the Randić index of trees, J. Math. Study 31, 225–230 (1998) (in Chinese)

    Google Scholar 

  162. B. Zelinka, Medians and peripherans of trees. Arch. Math. (Brno) 4, 87–95 (1968)

    MathSciNet  MATH  Google Scholar 

  163. X.D. Zhang, Y. Liu, M. Han, The maximum Wiener index of trees with given degree sequences. MATCH Commun. Math. Comput. Chem. 64, 661–682 (2010)

    MathSciNet  MATH  Google Scholar 

  164. X.D. Zhang, Q. Xiang, L. Xu, R. Pan, The Wiener index of trees with given degree sequences. MATCH Commun. Math. Comput. Chem. 60, 623–644 (2008)

    MathSciNet  MATH  Google Scholar 

  165. X.M. Zhang, X.D. Zhang, The minimal number of subtrees with a given degree sequence. Graphs Comb. 31(1), 309–318 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  166. X.M. Zhang, X.D. Zhang, D. Gray, H. Wang, Trees with the most subtrees—an algorithmic approach. J. Comb. 3(2), 207–223 (2012)

    MathSciNet  MATH  Google Scholar 

  167. X.M. Zhang, X.D. Zhang, D. Gray, H. Wang, The number of subtrees of trees with given degree sequence. J. Graph Theory 73(3), 280–295 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  168. J. Zito, The structure and maximum number of maximum independent sets in trees. J. Graph Theory 15(2), 207–221 (1991)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This research was supported in part by the NSF DMS contract 1300547 and by the DARPA and AFOSR under the contract FA9550-12-1-0405 (László Székely), National Research Foundation of South Africa, grant number 96236 (Stephan Wagner), and Simons Foundation grant 245307 (Hua Wang).

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Authors and Affiliations

  1. Department of Mathematics, University of South Carolina, Columbia, SC, USA

    László Székely

  2. Department of Mathematical Sciences, Stellenbosch University, Stellenbosch, South Africa

    Stephan Wagner

  3. Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA, USA

    Hua Wang

Authors
  1. László Székely
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  2. Stephan Wagner
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  3. Hua Wang
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Correspondence to László Székely .

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Editors and Affiliations

  1. Dept of Math, Statistics & Comp.Sc, Macalester College, St. Paul, Minnesota, USA

    Andrew Beveridge

  2. Dept. of Mathematics, University of South Carolina, Columbia, South Carolina, USA

    Jerrold R. Griggs

  3. Dept. of Mathematics, Iowa State University, Ames, Iowa, USA

    Leslie Hogben

  4. School of Mathematics, University of Minnesota, Minneapolis, Minnesota, USA

    Gregg Musiker

  5. School of Mathematics & Computer Sc, Georgia Institute of Technology, Atlanta, Georgia, USA

    Prasad Tetali

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Székely, L., Wagner, S., Wang, H. (2016). Problems related to graph indices in trees. In: Beveridge, A., Griggs, J., Hogben, L., Musiker, G., Tetali, P. (eds) Recent Trends in Combinatorics. The IMA Volumes in Mathematics and its Applications, vol 159. Springer, Cham. https://doi.org/10.1007/978-3-319-24298-9_1

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