图G的 Harmonic指标定义为图G 中所有边uv 的权重2du＋dv 的和，用 H（G）表示；二阶Harmonic指标被定义为图G中所有二长路uvw的权重3du＋dv＋dw 的和，用2H（G）表示；其中du表示G中点u的度数。该文研究了六角螺链和六角螺系统的 Harmonic 指标，发现它们的极图不唯一。通过研究二阶 Harmonic指标，确定了六角螺链的极图，发现六角螺系统的极图是一类特定的图。并且找到一个关于Harmonic 指标的极图和二阶 Harmonic 指标的极图的关系。最后，提出一个关于 Harmonic 指标的开放新问题。%The harmonic index and the second-order harmonic index of a graph G are defined as the sum of the weight 2du+dv of all edgesuv ofG and the sum of the weight 3 du+dv+dw of all 2-length paths uvw of G ,where du is the degree of a vertexu ofG ,denoted by H (G)and 2 H (G),respectively.In this paper,con-sidered the spiro hexagonal chains and the spiro hexagonal systems of the harmonic indices H (G),and seek out their extremal graphs were not unique.According to study the second-order harmonic index 2 H (G),they determined the extremal graphs of spiro hexagonal chains and found that the extremal graphs of spiro hexagon-al systems are a class of certain graphs.Finally,found a relation between the extremal graphs of H (G)and the extremal graphs of 2 H (G).A open problem on graph with harmonic index is proposed.
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