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2010•Journal of Applied and Industrial MathematicsRequires access

The wiener index of graphs of arbitrary girth and their line graphs

Andrey A. Dobrynin

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Abstract

We consider the invariant W ( G ) of a simple connected undirected graph G which is equal to the sum of distances between all pairs of its vertices in the natural metric (the Wiener index). We show that, for every g ≥ 5, there is a planar graph G of girth g satisfying W ( L ( G )) = W ( G ), where L ( G ) is the line graph of G .

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What this paper is about

We consider the invariant W ( G ) of a simple connected undirected graph G which is equal to the sum of distances between all pairs of its vertices in the natural metric (the Wiener index). We show that, for every g ≥ 5, there is a planar graph G of girth g satisfying W ( L ( G )) = W ( G ), where L ( G ) is the line graph of G .

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Available abstract

We consider the invariant W ( G ) of a simple connected undirected graph G which is equal to the sum of distances between all pairs of its vertices in the natural metric (the Wiener index). We show that, for every g ≥ 5, there is a planar graph G of girth g satisfying W ( L ( G )) = W ( G ), where L ( G ) is the line graph of G .

Key concepts: Mathematics, Combinatorics, Wiener index, Girth (graph theory), Discrete mathematics, Line graph, Graph

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