中文摘要：

图G的Merrifield-Simmons指标是指图G的独立集的个数,其中包括空集.文献[2]得到n阶单圈图中具有最大、次大、最小的Merrifield-Simmons指标的图类,以及讨论了当圈长为k时具有最大Merrifield-Simmons指标的图.文献[3]给出了圈长为k的n阶单圈图的第二大Merrifield-Simmons指标.本文得出了圈长为k的n阶单圈图的Merrifield-Simmons指标的第三大值和单圈图簇∪nk=3C（n,k）的Merrifield-Simmons指标的第三大值及对应的图.

英文摘要：

Let a-index, i. e., Merrifield-Simmons index, be the number of independent sets including the empty set. The unicyclic graphs of order n having the largest,second largest Merrifield-Simmons index were given in[2]. Moreover,the unicyclic graphs of order n and fen of g an gt raph G, smallest k having the largest, second largest Merrifield-Simmons index were investigated in [2] and [3],respectively. The third largest value of σ-index about uncyclic graphs of order n and length k and corresponding graphs were obtained.