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Article
Variations of Distance-Based Invariants of Trees
Journal of Combinatorial Mathematics and Combinatorial Computing
  • Marcus Bartlett, University of Georgia
  • Elliot Krop, Clayton State University
  • Colton Magnant, Georgia Southern University
  • Fedelis Mutiso, Georgia Southern University
  • Hua Wang, Georgia Southern University
Document Type
Article
Publication Date
11-1-2014
Disciplines
Abstract

Introduced in 1947, the Wiener index (sum of distances between all pairs of vertices) is one of the most studied chemical indices. Extensive results regarding the extremal structure of the Wiener index exist in the literature. More recently, the Gamma index (also called the Terminal Wiener index) was introduced as the sum of all distances between pairs of leaves. It is known that these two indices coincide in their extremal structures and that a nice functional relation exists for k-ary trees but not in general. In this note, we consider two natural extensions of these concepts, namely the sum of all distances between internal vertices (the Spinal index) and the sum of all distances between internal vertices and leaves (the Bartlett index). We first provide a characterization of the extremal trees of the Spinal index under various constraints. Then, its relation with the Wiener index and Gamma index is studied. The functional relation for k-ary trees also implies a similar result on the Bartlett index.

Citation Information
Marcus Bartlett, Elliot Krop, Colton Magnant, Fedelis Mutiso, et al.. "Variations of Distance-Based Invariants of Trees" Journal of Combinatorial Mathematics and Combinatorial Computing Vol. 91 (2014) p. 19 - 29 ISSN: 0835-3026
Available at: http://works.bepress.com/colton_magnant/45/