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2006 | 26 | 1 | 161-175
Tytuł artykułu

Wiener index of generalized stars and their quadratic line graphs

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The Wiener index, W, is the sum of distances between all pairs of vertices in a graph G. The quadratic line graph is defined as L(L(G)), where L(G) is the line graph of G. A generalized star S is a tree consisting of Δ ≥ 3 paths with the unique common endvertex. A relation between the Wiener index of S and of its quadratic graph is presented. It is shown that generalized stars having the property W(S) = W(L(L(S)) exist only for 4 ≤ Δ ≤ 6. Infinite families of generalized stars with this property are constructed.
Słowa kluczowe
Wydawca
Rocznik
Tom
26
Numer
1
Strony
161-175
Opis fizyczny
Daty
wydano
2006
otrzymano
2005-06-24
poprawiono
2005-07-22
Twórcy
  • Sobolev Institute of Mathematics, Russian Academy of Sciences, Siberian Branch, Novosibirsk 630090, Russia
  • Sobolev Institute of Mathematics, Russian Academy of Sciences, Siberian Branch, Novosibirsk 630090, Russia
Bibliografia
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  • [3] S.H. Bertz and W.F. Wright, The graph theory approach to synthetic analysis: definition and application of molecular complexity and synthetic complexity, Graph Theory Notes New York 35 (1998) 32-48.
  • [4] F. Buckley, Mean distance of line graphs, Congr. Numer. 32 (1981) 153-162.
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  • [6] Chemical Graph Theory - Introduction and Fundamentals, D. Bonchev and D.H. Rouvray, eds. (Gordon & Breach, New York, 1991).
  • [7] A.A. Dobrynin and I. Gutman, The Wiener index for trees and graphs of hexagonal systems, Diskretn. Anal. Issled. Oper. Ser. 2 5 (1998) 34-60, in Russian.
  • [8] A.A. Dobrynin, Distance of iterated line graphs, Graph Theory Notes New York 37 (1998) 8-9.
  • [9] A.A. Dobrynin, R. Entringer and I. Gutman, Wiener index for trees: theory and applications, Acta Appl. Math. 66 (2001) 211-249, doi: 10.1023/A:1010767517079.
  • [10] A.A. Dobrynin, I. Gutman, S. Klavžar and P. Zigert, Wiener index of hexagonal systems, Acta Appl. Math. 72 (2002) 247-294, doi: 10.1023/A:1016290123303.
  • [11] A.A. Dobrynin, I. Gutman and V. Jovasević, Bicyclic graphs and their line graphs with the same Wiener index, Diskretn. Anal. Issled. Oper. Ser. 2 4 (1997) 3-9, in Russian.
  • [12] A.A. Dobrynin and L.S. Mel'nikov, Wiener index for graphs and their line graphs with arbitrary large cyclomatic numbers, Appl. Math. Lett. 18 (2005) 307-312, doi: 10.1016/j.aml.2004.03.007.
  • [13] A.A. Dobrynin and L.S. Mel'nikov, Wiener index for graphs and their line graphs, Diskretn. Anal. Issled. Oper. Ser. 2 11 (2004) 25-44, in Russian.
  • [14] A.A. Dobrynin and L.S. Mel'nikov, Trees and their quadratic line graphs having the same Wiener index, MATCH Commun. Math. Comput. Chem. 50 (2004) 145-164.
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  • [16] A.A. Dobrynin and L.S. Mel'nikov, Wiener index, line graphs and the cyclomatic number, MATCH Commun. Math. Comput. Chem. 53 (2005) 209-214.
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  • [18] R.C. Entringer, Distance in graphs: trees, J. Combin. Math. Combin. Comput. 24 (1997) 65-84.
  • [19] I. Gutman, Distance of line graphs, Graph Theory Notes New York 31 (1996) 49-52.
  • [20] I. Gutman, V. Jovasević and A.A. Dobrynin, Smallest graphs for which the distance of the graph is equal to the distance of its line graph, Graph Theory Notes New York 33 (1997) 19.
  • [21] I. Gutman and L. Pavlović, More of distance of line graphs, Graph Theory Notes New York 33 (1997) 14-18.
  • [22] I. Gutman and O.E. Polansky, Mathematical Concepts in Organic Chemistry (Springer-Verlag, Berlin, 1986).
  • [23] I. Gutman, Y.N. Yeh, S.L. Lee and Y.L. Luo, Some recent results in the theory of the Wiener number, Indian J. Chem. 32A (1993) 651-661.
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  • [25] I. Gutman, L. Popović, B.K. Mishra, M. Kaunar, E. Estrada and N. Guevara, Application of line graphs in physical chemistry. Predicting surface tension of alkanes, J. Serb. Chem. Soc. 62 (1997) 1025-1029.
  • [26] F. Harary, Graph Theory, (Addison Wesley, 1969).
  • [27] G.H. Hardy, J.E. Littlewood and G. Polya, Inequalities (Cambridge University Press: Cambridge, 1934, 2nd ed. 1988).
  • [28] S. Nikolić, N. Trinajstić and Z. Mihalić, The Wiener index: developments and applications, Croat. Chem. Acta 68 (1995) 105-129.
  • [29] J. Plesnik, On the sum of all distances in a graph or digraph, J. Graph Theory 8 (1984) 1-21, doi: 10.1002/jgt.3190080102.
  • [30] O.E. Polansky and D. Bonchev, The Wiener number of graphs. I. General theory and changes due to some graph operations, MATCH Commun. Math. Comput. Chem. 21 (1986) 133-186.
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  • [32] N. Trinajstić, Chemical Graph Theory (CRC Press: Boca Raton, 1983; 2nd ed. 1992).
  • [33] H. Wiener, Structural determination of paraffin boiling points, J. Amer. Chem. Soc. 69 (1947) 17-20, doi: 10.1021/ja01193a005.
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1310
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