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It is a classical result that any permutation in the symmetric group can begenerated by a sequence of adjacent transpositions. The sequences of minimallength are called reduced words, and in this paper we study the graphs of thesereduced words, with edges determined by relations in the underlying Coxetergroup. Recently, the diameter has been calculated for the longest permutation$n\ldots 21$ by Reiner and Roichman as well as Assaf. In this paper we findinductive formulas for the diameter of the graphs of 12-inflations and many21-inflations. These results extend to the associated graphs on commutation andlong braid classes. Also, these results give a recursive formula for thediameter of the longest permutation, which matches that of Reiner, Roichman andAssaf. Lastly, We make progress on conjectured bounds of the diameter by Reinerand Roichman, which are based on the underlying hyperplane arrangement, andfind families of permutations that achieve the upper bound and lower bound ofthe conjecture. In particular permutations that avoid 312 or 231 have graphsthat achieve the upper bound.

Diameters of Graphs on Reduced Words of 12 and 21-Inflations