On the Augmented Sombor Index of Graphs

Abstract

Let $G$ be a connected graph having more than two vertices and let $d_i$ denote the degree of vertex $v_i$ in $G$. Let $E(G)$ represent the edge set of $G$. Then, the augmented Sombor (ASO) index of $G$ is defined as $ASO(G)={\sum }_{v_i{v}_j\in E(G)}\sqrt{(d_i+d_j-2{)}^{-1}(d_i^2+d_j^2)}.$ It is known that the cycle graph $C_n$ uniquely minimizes the ASO index in the class of all $n$-order unicyclic graphs. In this paper, we prove that the unique $n$-order unicyclic graph of maximum degree $n-1$ maximizes the ASO index in the aforementioned unicyclic graph class. We also prove that $ASO(G-v_i{v}_j)<ASO(G)$ whenever neither of the graphs $G-v_i{v}_j$ and $G$ contains any isolated edge. Utilizing this edge-deletion property, we characterize the unique graph maximizing the ASO index among all fixed-order connected graphs with a specified vertex connectivity (or edge connectivity).

24 pages