Wiener index of generalized stars and their quadratic line graphs

Andrey A. Dobrynin ; Leonid S. Mel'nikov

Discussiones Mathematicae Graph Theory, Tome 26 (2006), p. 161-175 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 2006-01-01

Zbl 1102.05019

EUDML-ID : urn:eudml:doc:270582

Bibliographie

[000] [1] A.T. Balaban, I. Motoc, D. Bonchev and O. Mekenyan, Topological indices for structure-activity correlations, Topics Curr. Chem. 114 (1983) 21-55, doi: 10.1007/BFb0111212.

[001] [2] S.H. Bertz, Branching in graphs and molecules, Discrete Appl. Math. 19 (1988) 65-83, doi: 10.1016/0166-218X(88)90006-6. | Zbl 0633.05063

[002] [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.

[003] [4] F. Buckley, Mean distance of line graphs, Congr. Numer. 32 (1981) 153-162.

[004] [5] E.R. Canfield, R.W. Robinson and D.H. Rouvray, Determination of the Wiener molecular branching index for the general tree, J. Comput. Chem. 6 (1985) 598-609, doi: 10.1002/jcc.540060613.

[005] [6] Chemical Graph Theory - Introduction and Fundamentals, D. Bonchev and D.H. Rouvray, eds. (Gordon & Breach, New York, 1991).

[006] [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. | Zbl 0913.05042

[007] [8] A.A. Dobrynin, Distance of iterated line graphs, Graph Theory Notes New York 37 (1998) 8-9.

[008] [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. | Zbl 0982.05044

[009] [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. | Zbl 0993.05059

[010] [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. | Zbl 0935.05040

[011] [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. | Zbl 1066.05060

[012] [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. | Zbl 1079.05031

[013] [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. | Zbl 1052.05029

[014] [15] A.A. Dobrynin and L.S. Mel'nikov, Trees, quadratic line graphs and the Wiener index, Croat. Chem Acta 77 (2004) 477-480.

[015] [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. | Zbl 1080.05028

[016] [17] R.C. Entringer, D.E. Jackson and D.A. Snyder, Distance in graphs, Czechoslovak Math. J. 26 (1976) 283-296. | Zbl 0329.05112

[017] [18] R.C. Entringer, Distance in graphs: trees, J. Combin. Math. Combin. Comput. 24 (1997) 65-84. | Zbl 0880.05029

[018] [19] I. Gutman, Distance of line graphs, Graph Theory Notes New York 31 (1996) 49-52.

[019] [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.

[020] [21] I. Gutman and L. Pavlović, More of distance of line graphs, Graph Theory Notes New York 33 (1997) 14-18.

[021] [22] I. Gutman and O.E. Polansky, Mathematical Concepts in Organic Chemistry (Springer-Verlag, Berlin, 1986). | Zbl 0657.92024

[022] [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.

[023] [24] I. Gutman and E. Estrada, Topological indices based on the line graph of the molecular graph, J. Chem. Inf. Comput. Sci. 36 (1996) 541-543, doi: 10.1021/ci950143i.

[024] [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.

[025] [26] F. Harary, Graph Theory, (Addison Wesley, 1969).

[026] [27] G.H. Hardy, J.E. Littlewood and G. Polya, Inequalities (Cambridge University Press: Cambridge, 1934, 2nd ed. 1988). | Zbl 0010.10703

[027] [28] S. Nikolić, N. Trinajstić and Z. Mihalić, The Wiener index: developments and applications, Croat. Chem. Acta 68 (1995) 105-129.

[028] [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. | Zbl 0552.05048

[029] [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. | Zbl 0603.05045

[030] [31] D.H. Rouvray, Should we have designs on topological indices?, in: R.B. King, ed., Chemical Application of Topology and Graph Theory, (Elsevier, Amsterdam, 1983) 159-177.

[031] [32] N. Trinajstić, Chemical Graph Theory (CRC Press: Boca Raton, 1983; 2nd ed. 1992).

[032] [33] H. Wiener, Structural determination of paraffin boiling points, J. Amer. Chem. Soc. 69 (1947) 17-20, doi: 10.1021/ja01193a005.