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Eccentric connectivity index of hexagonal belts and chains. (English) Zbl 1289.05117
The eccentric connectivity index (for short, ECI) of a simple graph $$G$$ is defined as $$\sum_{v\in V(G)}d(v)\operatorname{ecc}(v)$$, where $$d(v)$$ and $$\operatorname{ecc}(v)$$ denote degree and eccentricity of the corresponding vertex $$v$$, respectively. The relevance of this invariant is recognized in a number of biological and chemical papers.
The authors give formulae for computing the ECI of zigzag and armchair hexagonal belts (the graphs with chemical structure consisting of hexagonal facets where two adjacent facets share an edge). The formulae for open chains of the same belts are also given.
Although the results are purely mathematical, the classes of graphs considered are very specific to chemistry.

##### MSC:
 05C12 Distance in graphs 05C40 Connectivity 05C90 Applications of graph theory 92E10 Molecular structure (graph-theoretic methods, methods of differential topology, etc.)
##### Keywords:
chemical graphs; graph distances; eccentricity