@

초록

The atom-bond sum-connectivity (ABS) index, introduced in 2022, was formulated by integrating key concepts from two well-established topological indices. For α∈R, the general atom-bond sum-connectivity index of a graph G is defined as ABSα(G)=∑uv∈E(G)du+dv−2du+dvα, where E(G) is edge set of G, and du and dv are the degrees of vertices u and v, respectively. Ali et al. put forward two conjectures on the extremal graphs with minimum ABS1/2(G) among all bicyclic graphs and tricyclic graphs (Ali et al. (2024)). The conjecture on bicyclic graphs was later fully resolved independently by Das (2026) and Qiu et al. (2024), and the conjecture on tricyclic graphs was later fully resolved by Das (0000). In this paper, we show that the extremal graphs with minimum ABSα index among all c-cyclic graphs with n vertices contains at most one pendant vertex for c≥1 and 1/2≤α≤1. Based on this, we also determine the minimum ABSα index for bicyclic graphs and tricyclic with n vertices, where 1/2≤α≤1. Our results extend the main results of Das (2026, 0000) and Qiu et al. (2024).

키워드