A note on the maximal inverse sum indeg index of trees

The inverse sum indeg index (ISI index) of a graph G is defined as, where d(vi) is the degree of a vertex vi.It is known that the star Sn uniquely minimizes the ISI index among trees of order n.However, characterizing trees of order n with the maximal ISI index (optimal trees, for convenience) appears to be difficult.Chen, Li, and Lin in Appl.Math.Comput.392 (2021) #125731 gave some structural properties and three conjectures regarding an optimal tree.In this paper, the trees within a set T Sn of trees of order n are investigated, where T Sn is defined in the main text and it is the set to which the optimal tree is conjectured to belong.Several structural properties associated with an optimal tree are presented.The findings of the present paper imply that if the second part of Conjecture 4.3 of the mentioned paper holds, then its remaining two conjectures are also valid.