拟单圈图的调和指数

On harmonic index of quasi-unicyclic graphs

调和指数是一个与图的边和顶点度的相关概念,调和指数在图中代表了一种度量图的边权重的方式.设图G是n阶的简单图,图G的调和指数H(G)定义为H(G)=∑_(uv∈E(G))2/d(u)+d(v),其中E(G)表示图G中的边,d(u)和d(v)分别在图G中表示顶点u和v的度.拟单圈图是一类特殊的图,它不是单圈图,且在图G中存在点u∈V(G),使得G-u为连通的单圈图,则图G就称为拟单圈图.针对d(u)≥2的情况下,给出了拟单圈图的调和指数的下界,并在此情况下刻画了极图.

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 vertex 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.