On the Augmented Sombor Index of Graphs

TL;DR

This work investigates the extremal behavior of the augmented Sombor index (ASO) for graphs, focusing on unicyclic graphs and connected graphs with prescribed vertex or edge connectivity. It develops an edge-deletion principle and leverages degree-based bounds to derive sharp extremal results. The authors prove that the unique $n$-vertex unicyclic graph with maximum degree $n-1$ maximizes ASO, and they characterize the unique extremal connected graphs of order $n$ with given connectivity as $(K_1\cup K_{n-k-1})\vee K_k$ (with $K_n$ arising when $k=n-1$). The results advance understanding of ASO's extremal structure and suggest further exploration across more graph families.

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).

Theorems & Definitions (14)

• Lemma 1
• proof
• Lemma 2
• Theorem 1
• proof
• Claim 1
• Lemma 3
• proof
• Theorem 2
• proof