The Minimum Harmonic Index for Unicyclic Graphs with Given Diameter

Abstract

The harmonic index of a graph G is defined as the sum of the weights 2d(u)+d(v)${2 \over {d(u) + d(v)}}$ 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
• 05c07
• 05c35
• 92e10