Wiener index of iterated line graphs of trees homeomorphic to

Abstract This is fourth paper out of five in which we completely solve a problem of Dobrynin, Entringer and Gutman. Let G be a graph. Denote by L i ( G ) its i -iterated line graph and denote by W ( G ) its Wiener index. Moreover, denote by H a tree on six vertices, out of which two have degree 3 and four have degree 1. Let j ≥ 3 . In previous papers we proved that for every tree T , which is not homeomorphic to a path, claw K 1 , 3 and H , it holds W ( L j ( T ) ) > W ( T ) . Here we prove that W ( L 4 ( T ) ) > W ( T ) for every tree T homeomorphic to H . As a consequence, we obtain that with the exception of paths and the claw K 1 , 3 , for every tree T it holds W ( L i ( T ) ) > W ( T ) whenever i ≥ 4 .

[1]  Frank Harary,et al.  Distance in graphs , 1990 .

[2]  Ivan Gutman,et al.  Wiener Index and Vibrational Energy , 2002 .

[3]  Martin Knor,et al.  On a conjecture about Wiener index in iterated line graphs of trees , 2012, Discret. Math..

[4]  Baoyindureng Wu,et al.  Wiener Index of Line Graphs , 2010 .

[5]  Yong Liu,et al.  Maximum Wiener Index of Trees with Given Degree Sequence , 2010 .

[6]  Bo Zhou,et al.  MINIMUM WIENER INDICES OF TREES AND UNICYCLIC GRAPHS OF GIVEN MATCHING NUMBER , 2010 .

[7]  Ján Plesník,et al.  On the sum of all distances in a graph or digraph , 1984, J. Graph Theory.

[8]  Hong Bian,et al.  The Polyphenyl Chains with Extremal Edge-Wiener Indices ∗ , 2010 .

[9]  Ali Iranmanesh,et al.  COMPUTATION OF THE FIRST EDGE-WIENER INDEX OF TUC4C8(S) NANOTUBE , 2009 .

[10]  Bo Zhou,et al.  Minimum sum-connectivity indices of trees and unicyclic graphs of a given matching number , 2010 .

[11]  Martin Knor,et al.  The Wiener index in iterated line graphs , 2012, Discret. Appl. Math..

[12]  Peter Dankelmann,et al.  The edge-Wiener index of a graph , 2009, Discret. Math..

[13]  I. Gutman,et al.  Wiener Index of Trees: Theory and Applications , 2001 .

[14]  I. Gutman,et al.  Wiener Index of Hexagonal Systems , 2002 .

[15]  Bruce S. Elenbogen,et al.  Distance distributions for graphs modeling computer networks , 2007, Discret. Appl. Math..

[16]  H. Wiener.  Structural determination of paraffin boiling points. , 1947, Journal of the American Chemical Society.

[17]  Roi Krakovski,et al.  On Wiener index of graphs and their line graphs , 2010 .

[18]  Tomaz Pisanski,et al.  Edge-contributions of some topological indices and arboreality of molecular graphs , 2009, Ars Math. Contemp..

[19]  Wei Luo,et al.  On ordinary and reverse Wiener indices of non-caterpillars , 2009, Math. Comput. Model..

[20]  Bolian Liu,et al.  On the variable Wiener indices of trees with given maximum degree , 2010, Math. Comput. Model..

[21]  Martin Knor,et al.  Wiener index of iterated line graphs of trees homeomorphic to the claw K1, 3 , 2013, Ars Math. Contemp..