Geodesic
The Minimum Harmonic Index for Unicyclic Graphs with Given Diameter
Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 2, pp. 429-442

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The harmonic index of a graph G is defined as the sum of the weights 2d(u)+d(v) of all edges uv of G, where d(u) denotes the degree of a vertex u in G. In this paper, we present the minimum harmonic index for unicyclic graphs with given diameter and characterize the corresponding extremal graphs. This answers an unsolved problem of Zhu and Chang [26].
Keywords: harmonic index, unicyclic graphs, diameter 
Zhong, Lingping. The Minimum Harmonic Index for Unicyclic Graphs with Given Diameter. Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 2, pp. 429-442. http://geodesic.mathdoc.fr/item/DMGT_2018_38_2_a6/
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