The $k$-apex trees with minimum augmented Zagreb index

TL;DR

This paper identifies the $k$-apex trees with the minimum augmented Zagreb index among all such graphs for certain parameters, solving an open problem in graph theory related to topological indices.

Contribution

It determines the structure of $k$-apex trees that minimize the AZI, addressing an open problem in the characterization of extremal graphs for this index.

Findings

Graphs minimizing AZI among $k$-apex trees are characterized for $k extgreater=4$ and $n extgreater=3(k+1)$.

The study provides structural properties of $k$-apex trees relevant to AZI minimization.

The results solve an open problem from previous literature on topological indices in graph theory.

Abstract

For a connected graph $G$ on at least three vertices, the augmented Zagreb index (AZI) of $G$ is defined as $$AZI(G)=\sum_{uv\in E(G)}\left(\frac{d(u)d(v)}{d(u)+d(v)-2}\right)^{3},$$ being a topological index well-correlated with the formation heat of heptanes and octanes. A $k$-apex tree $G$ is a connected graph admitting a $k$-subset $X\subset V(G)$ such that $G-X$ is a tree, while $G-S$ is not a tree for any $S\subset V(G)$ of cardinality less than $k$. By investigating some structural properties of $k$-apex trees, we identify the graphs minimizing the AZI among all $k$-apex trees on $n$ vertices for $k\ge 4$ and $n\ge 3(k+1)$. The latter solves an open problem posed in [K. Cheng, M. Liu, F. Belardo, {\em Appl. Math. Comput.}, {\bf402} (2021), 126139].