On the parity of the Wiener index

Authors:
Stephan Wagner;Hua Wang
Affiliations:
Department of Mathematical Sciences, Stellenbosch University, Stellenbosch 7602, South Africa;Mathematical Sciences, Georgia Southern University, Statesboro, GA, 30460, United States
Venue:
European Journal of Combinatorics
Year:
2009

Citing 2
Cited 1

Parity reversing involutions on plane trees and 2-Motzkin paths

European Journal of Combinatorics
Analytic Combinatorics

Analytic Combinatorics

Note: Wiener index versus Szeged index in networks

Discrete Applied Mathematics

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Abstract

It is a known fact that the Wiener index (i.e. the sum of all distances between pairs of vertices in a graph) of a tree with an odd number of vertices is always even. In this paper, we consider the distribution of the Wiener index and the related tree parameter ''internal path length'' modulo 2 by means of a generating functions approach as well as by constructing bijections for plane trees.