Abstract: Firstly, we solved the problem of the maximum, second-maximum and second-minimum values of the reduced Sombor index 
of n(n≥5) order  unicyclic graphs, and the corresponding extremal graphs by using  graphical transformations and unique classification methods. Secondly, we considered the problems of the second-maximum and second-minimum values of the reduced Sombor index  of  n(n≥6) order bicyclic graphs, and the corresponding extremal graphs, gave the maximum, second-maximum 
and second-minimum values of the reduced Sombor index  of  n(n≥5) order  unicyclic graphs, and characterized the corresponding extremal graphs. At the same time, we also confirmed the second-maximum and second-minimum  values of the reduced Sombor index  of  n(n≥6) order bicyclic graphs, and the corresponding extremal graphs.

Key words: reduced Sombor index, unicyclic graph, bicyclic graph, extremal graph