Maximal tree and unicylic graph for Euler Sombor index with given diameter

The study of topological descriptors is essential for understanding the underlying structures of graphs and networks. Numerous numerical descriptors associated with graphs have been used to analyze their overall structure. In this analysis, degree-based topological descriptors hold a significant place. The Euler Sombor index of a graph [Formula: see text] is a significant vertex degree-based topological index related to the Sombor index, showing a strong correlation with the physicochemical properties of octanes. It represents the perimeter of an ellipse, with focal points recognized as the degree-point and the dual-point of a pair of connected vertices in [Formula: see text]. Nowadays, finding extremal results with respect to various graph indices for fixed graph parameters has become an important and engaging area of research in extremal graph theory. In this article, we explore the first and second maximum Euler Sombor index of a tree with a fixed diameter [Formula: see text]. The extremal trees are also identified. Furthermore, we provide the ordering of trees for [Formula: see text]. In addition, we identify the maximal unicyclic graph when the graph order and diameter are given.