Extremal unicyclic and bicyclic graphs of the Euler Sombor index

Topological indices are widely used to analyze and predict the physicochemical properties of compounds, and have good application prospects. Recently, the Euler Sombor index was introduced, which is defined as

$$\begin{array}{rl}EP(G)=\sum _{v_i{v}_j\in E(G)}\sqrt{d(v_i{)}^2+d(v_j{)}^2+d(v_i)d(v_j)}.& \end{array}$$

As the latest index with geometry motivation, it has excellent discrimination and predictive ability for compounds, in addition to mathematical practicality. The unicyclic graphs and bicyclic graphs are composed of various chemical structures, and are of particular importance in the study of topological indices. In this paper, the maximal and minimal values of Euler Sombor index for all unicyclic and bicyclic graphs are determined, and the corresponding extremal graphs are characterized.