Some New Inequalities Concerning the Variable Sum Exdeg Index/Coindex of Graphs

Abstract

The variable sum exdeg index and coindex of a graph $G$ are denoted by $SEI_a(G)$ and $\overline{SEI}_a(G)$, respectively, and they are defined as
$SEI_a(G)=\sum_{i=1}^n d_i a^{d_i}$ and $\overline{SEI}_a(G)=\sum_{i=1}^n (n-1-d_i) a^{d_i}$, respectively, where `$a$' is a positive real
number different from $1$ and $(d_1,d_2,\cdots, d_n)$ is the vertex-degree sequence of $G$.
The present paper gives several new inequalities involving the graph invariants $SEI_a$\, and/or \,$\overline{SEI}_a$. All graphs attaining the equality signs in the obtained inequalities are also characterized.