Abstract
The augmented Zagreb index of a graph G is defined as $$ \operatorname{AZI}(G)=\sum_{uv\in E(G)} \biggl( \frac{d_{u}d_{v}}{d_{u}+d_{v}-2} \biggr)^{3}, $$ where $E(G)$ is the edge set, and $d_{u}$ , $d_{v}$ are the degrees of vertices u and v in G, respectively. This new molecular structure descriptor, introduced by Furtula et al. (J. Math. Chem. 48:370-380, 2010), has proven to be a valuable predictive index in the study of the heat of formation in heptanes and octanes. In this paper, the n-vertex unicyclic graphs with the minimal and the second minimal AZI indices and the n-vertex bicyclic graphs with the minimal AZI index are determined.
Highlights
Let G = (V, E) be a simple, finite and undirected graph of order n = |V | and size m = |E|
The maximum vertex degree is denoted by, the minimum vertex degree is denoted by δ, and the minimum non-pendent vertex degree is denoted by δ
If u and v are two adjacent vertices of G, the edge connecting them will be denoted by uv [ ]
Summary
In , Furtula et al [ ] proposed a new, vertex-degree-based graph topological index called the augmented Zagreb index (AZI), defined as. Gutman and Tošovič [ ] recently tested the correlation abilities of vertex-degree-based topological indices for the case of standard heats of formation and normal boiling points of octane isomers. They found that the AZI index yields the best results. Ali et al [ ] established inequalities between AZI and several other vertex-degree-based topological indices. Ali et al [ ] proposed tight upper bounds for the AZI of chemical bicyclic and unicyclic graphs and provided a Nordhaus-Gaddum-type result for the AZI index. The n-vertex bicyclic graphs in which the AZI index attains its minimal value are obtained
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