Trees With Given Degree Sequence in S-Order

In this note we consider the lexicographical ordering by spectral moments of trees with given degree sequence. Such questions have been studied for a variety of different categories of trees. Particularly, the last tree in this ordering among trees with given degree sequence was recently identified in two independent manuscripts. The characterization of the first such trees, however, remains open. We make some progress on this question in this note, by making use of the interpretation of the spectral moment in terms of numbers of paths and the product of adjacent vertex degrees, the first trees are characterized with the additional condition that the nonleaf vertex degrees are different from each other. We also comment on the case when there are repetitions in the vertex degrees.