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Discrete Applied Mathematics

Volume 181, 30 January 2015, Pages 167-173

The connectivity and the Harary index of a graph

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Abstract

The Harary index of a graph is defined as the sum of reciprocals of distances between all pairs of vertices of the graph. In this paper we provide an upper bound of the Harary index in terms of the vertex or edge connectivity of a graph. We characterize the unique graph with the maximum Harary index among all graphs with a given number of cut vertices or vertex connectivity or edge connectivity. In addition we also characterize the extremal graphs with the second maximum Harary index among all graphs with given vertex connectivity.

Keywords

Graph
Harary index
Cut vertex
Vertex connectivity
Edge connectivity

This work was supported by National Natural Science Foundation of China (11071002, 11371028), Program for New Century Excellent Talents in University (NCET-10-0001), Key Project of Chinese Ministry of Education (210091), Specialized Research Fund for the Doctoral Program of Higher Education (20103401110002), Natural Science Research Foundation of Department of Education of Anhui Province (KJ2013A196), Science and Technological Fund of Anhui Province for Outstanding Youth (10040606Y33), Scientific Research Fund for Fostering Distinguished Young Scholars of Anhui University (KJJQ1001kok体育官方网站), Academic Innovation Team of Anhui University Project (KJTD001B).

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