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On eccentric connectivity index and connectivity. (English) Zbl 1415.05090

Summary: Let \(G\) be a finite connected graph. The eccentric connectivity index \({\xi}^c(G)\) of \(G\) is defined as \(\xi^c (G)) = \Sigma_{V \in V (G)} \operatorname{ec}(v)) \deg(v)\), where \(\operatorname{ec}(v)\) and \(\deg(v)\) denote the eccentricity and degree of a vertex \(v\) in \(G\), respectively. In this paper, we give an asymptotically sharp upper bound on the eccentric connectivity index in terms of order and vertex-connectivity and in terms of order and edge-connectivity. We also improve the bounds for triangle-free graphs.

MSC:

05C35 Extremal problems in graph theory
05C12 Distance in graphs
05C40 Connectivity
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References:

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