On the variable Wiener indices of trees with given maximum degree

  • Authors:

  • Muhuo Liu;Bolian Liu

  • Affiliations:

  • Department of Applied Mathematics, South China Agricultural University, Guangzhou, 510642, PR China and School of Mathematic Science, South China Normal University, Guangzhou, 510631, PR China;School of Mathematic Science, South China Normal University, Guangzhou, 510631, PR China

  • Venue:
  • Mathematical and Computer Modelling: An International Journal
  • Year:
  • 2010

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Abstract

The Wiener index of a tree T obeys the relation W(T)=@?"en"1(e)@?n"2(e), where n"1(e) and n"2(e) are the number of vertices adjacent to each of the two end vertices of the edge e, respectively, and where the summation goes over all edges of T. Lately, Nikolic, Trinajstic and Randic put forward a novel modification ^mW of the Wiener index, defined as ^mW(T)=@?"e(n"1(e)@?n"2(e))^-^1. Very recently, Gutman, Vukic@?evic and Z@?erovnik extended the definitions of W(T) and ^mW(T) to be ^mW"@l(T)=@?"e(n"1(e)@?n"2(e))^@l, and they called ^mW the modified Wiener index of T, and ^mW"@l(T) the variable Wiener index of T. Let @D(T) denote the maximum degree of T. Let T"n denote the set of trees on n vertices, and T"n^c={T@?T"n|@D(T)=c}. In this paper, we determine the first two largest (resp. smallest) values of ^mW"@l(T) for @l0 (resp. @l=n2. And we identify the first two largest and first three smallest Wiener indices in T"n^c(c=n2), respectively. Moreover, the first two largest and first two smallest modified Wiener indices in T"n^c(c=n2) are also identified, respectively.