The harmonic index is a concept related to the edges and vertex degrees of a graph.It represents a way to measure the edge weights in a graph.Let G be a simple graph of order n.The harmonic index H(G)of graph G was defined as H(G)=∑ uv∈E(G)2/d(u)+d(v),where E(G)denotes the edges in graph G,and d(u)and d(v)represent the degrees of ver-tex u and v in graph G,respectively.A quasi-unicyclic graph is a special type of graph that was not a unicyclic graph but contains a vertex u∈V(G)such that G-u was a connected unicyclic graph.This paper explored and provided a lower bound for the harmonic index of quasi-unicyclic graphs under the condition that d(u)≥2,and characterized the extremal graphs that achieve this bound.
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