On the Augmented Sombor Index of Graphs

TL;DR

This paper investigates the augmented Sombor index of graphs, proving extremal properties for unicyclic graphs and characterizing graphs with maximum ASO index under connectivity constraints.

Contribution

It establishes the maximum ASO index for unicyclic graphs with maximum degree n-1 and characterizes graphs with maximum ASO index given fixed order and connectivity.

Findings

Cycle graph minimizes ASO in unicyclic graphs.

Unicyclic graph with maximum degree n-1 maximizes ASO.

Edge deletion decreases ASO unless isolated edges are involved.

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 $ASO(G-v_iv_j)<ASO(G)$ whenever neither of the graphs $G-v_iv_j$ 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).