On diameter-constrained trees and BID indices

Let G denote a tree with edge set E. For a vertex w of G, its degree is denoted as. This study investigates graph invariants of the type, where ϑ is a symmetric real-valued function based on the degrees of adjacent vertices in G. Such graph invariants are often referred to as bond incident degree (BID) indices. The primary objective is to identify trees that either minimize or maximize B_ϑ in the set of trees of a fixed order and a predetermined diameter, under explicit conditions involving the function ϑ. These conditions are satisfied by many indices, thereby granting the obtained findings broad applicability across numerous classical and contemporary BID indices. A key outcome includes the precise characterization of graphs that maximize many particular BID indices, such as the atom-bond sum-connectivity, Sombor, Euler–Sombor, elliptic Sombor, and Zagreb–Sombor indices, within the aforementioned set of trees. Some obtained results also facilitate the characterization of trees that minimize various other indices in the aforesaid set of trees, including the harmonic and sum-connectivity indices. The paper concludes with a conjecture based on computations carried out using software for trees of order up to 15.