Maximal Augmented Zagreb Index of Trees With at Most Three Branching Vertices

The Augmented Zagreb index of a graph $G$ is defined to be $\mathcal {AZI}\left ({G}\right) =\sum _{uv\in E\left ({G}\right) } \left ({\frac { d\left ({u}\right) d\left ({v}\right) }{d\left ({u}\right) +d\left ({v}\right) -2} }\right) ^{3}$ , where $E\left ({G}\right) $ is the edge set of $G$ , $d\left ({u}\right) $ and $d\left ({v}\right) $ are the degrees of the vertices $u$ and $v$ of edge $uv$ . It is one of the most valuable topological indices used to predict the structure-property correlations of organic compounds. It is well known that the star is the unique tree having minimal $\mathcal {AZI}$ among trees. However, the problem of finding the tree with maximal $\mathcal {AZI}$ is still open and seems to be a very difficult problem. A recent conjecture, posed in the recent paper [IEEE Access, vol. 6, pp. 69335–69341, 2018], states that the balanced double star is the tree with maximal $\mathcal {AZI}$ among all trees with $n$ vertices, for all $n\geq 19$ . Let $\Omega \left ({n,p}\right) $ be the set of trees with $n$ vertices and $p$ branching vertices. In this paper we consider the maximal value problem of $\mathcal {AZI}$ over $\Omega \left ({n,p}\right) $ . We first show that under a certain condition, the problem reduces to finding the maximal value of $\mathcal {AZI}$ over $\Omega _{1}\left ({n,p}\right) $ , the set of trees in $\Omega \left ({n,p}\right) $ with no vertices of degree 2. Then we rely on this result to find the trees with maximal value of $\mathcal {AZI}$ over $\Omega \left ({n,p}\right) $ , when $p=2$ and 3. In particular, we deduce that the conjecture holds for all trees with at most 3 branching vertices.