Graphs that minimizing max–min rodeg index

The max–min rodeg ( $${M\!m_{sde}}$$ ) index is a useful topological index in mathematical chemistry. Damir Vuki $$\check{\text {c}}$$ evi $$\acute{\text {c}}$$ studied the mathematical properties of the max–min rodeg index. In this paper, we determine the n-vertex trees with the second, the third and the fourth for $$n\ge 7$$ , and the fifth for $$n\ge 10$$ minimum $$M\!m_{sde}$$ indices, unicyclic graphs with the second and the third for $$n\ge 5$$ , the fourth, the fifth and the sixth for $$n\ge 7$$ , and the seventh for $$n\ge 9$$ minimum $$M\!m_{sde}$$ indices, and bicyclic graphs with the first for $$n\ge 4$$ , the second and the third for $$n\ge 6$$ , and the fourth for $$n\ge 8$$ minimum $$M\!m_{sde}$$ indices.