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The Connectivity and the Harary Index of a Graph
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 maximum Harary index among all
graphs with 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 the graphs with given vertex connectivity
Cacti with Extremal PI Index
The vertex PI index is a
distance-based molecular structure descriptor, where denotes the
number of vertices which are closer to the vertex than to the vertex
and which has been the considerable research in computational chemistry dating
back to Harold Wiener in 1947. A connected graph is a cactus if any two of its
cycles have at most one common vertex. In this paper, we completely determine
the extremal graphs with the largest and smallest vertex PI indices among all
the cacti. As a consequence, we obtain the sharp bounds with corresponding
extremal cacti and extend a known result.Comment: Accepted by Transactions on Combinatorics, 201
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