TY - JOUR

T1 - Variations of Distance-Based Invariants of Trees

AU - Bartlett, Marcus

AU - Krop, Elliot

AU - Magnant, Colton

AU - Mutiso, Fedelis

AU - Wang, Hua

PY - 2014/11/1

Y1 - 2014/11/1

N2 - 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.

AB - 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.

KW - Combinatorial Computing

KW - Combinatorial Mathematics

KW - Distance-based invariants

KW - Variations

UR - https://digitalcommons.georgiasouthern.edu/math-sci-facpubs/333

UR - https://www.researchgate.net/publication/288175719_Variations_of_distance-based_invariants_of_trees

UR - http://www.scopus.com/inward/record.url?scp=84914677492&partnerID=8YFLogxK

M3 - Article

SN - 0835-3026

VL - 91

SP - 19

EP - 29

JO - Journal of Combinatorial Mathematics and Combinatorial Computing

JF - Journal of Combinatorial Mathematics and Combinatorial Computing

ER -