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Eccentric connectivity index and eccentric distance sum of some graph operations. (English) Zbl 1319.05082
The eccentric connectivity index (ECI) of a graph \(G=(V,E)\) is the sum over all vertices \(v\in V\) of the product of the degree and the eccentricity of \(v\). The eccentric distance sum (EDS) is defined analogously, just replace the degree of \(v\) with the total distance of \(v\). The generalized hierarchical product of two graphs is obtained from their Cartesian product by removing all the edges from selected layers of the second factor. In this paper, the ECI and the EDS of a generalized hierarchical product graph are expressed in terms of related invariants of the factors. Several applications are given, notably the ECI of four so-called \(F\)-sum graphs are obtained.

05C40 Connectivity
05C12 Distance in graphs
05C76 Graph operations (line graphs, products, etc.)