On the First Three Extremum Values of Variable Sum Exdeg Index of Trees

Abstract

For a graph $G$ , its variable sum exdeg index is defined as ${\text{SEI}}_a\left(G\right)={\displaystyle {\sum }_{xy\in E\left(G\right)}\left[a^{d_x}+a^{d_y}\right]}$ , where $a$ is a real number other than 1 and $d_x$ is the degree of a vertex $x$ . In this paper, we characterize all trees on $n$ vertices with first three maximum and first three minimum values of the ${\text{SEI}}_a$ index. Also, we determine all the trees of order $n$ with given diameter $d$ and having first three largest values of the ${\text{SEI}}_a$ index.