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The Augmented Zagreb index of a graph <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> is defined to be <inline-formula> <tex-math notation="LaTeX">$\mathcal {AZI}\left ({G}\right) =\sum _{uv\in E\left ({G}\right) } \left ({\frac { d\left ({u}\right) d\left ({v}\right) }{d\left ({u}\right) +d\left ({v}\right) -2} }\right) ^{3}$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$E\left ({G}\right) $ </tex-math></inline-formula> is the edge set of <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$d\left ({u}\right) $ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$d\left ({v}\right) $ </tex-math></inline-formula> are the degrees of the vertices <inline-formula> <tex-math notation="LaTeX">$u$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$v$ </tex-math></inline-formula> of edge <inline-formula> <tex-math notation="LaTeX">$uv$ </tex-math></inline-formula>. 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 <inline-formula> <tex-math notation="LaTeX">$\mathcal {AZI}$ </tex-math></inline-formula> among trees. However, the problem of finding the tree with maximal <inline-formula> <tex-math notation="LaTeX">$\mathcal {AZI}$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$\mathcal {AZI}$ </tex-math></inline-formula> among all trees with <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> vertices, for all <inline-formula> <tex-math notation="LaTeX">$n\geq 19$ </tex-math></inline-formula>. Let <inline-formula> <tex-math notation="LaTeX">$\Omega \left ({n,p}\right) $ </tex-math></inline-formula> be the set of trees with <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> vertices and <inline-formula> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> branching vertices. In this paper we consider the maximal value problem of <inline-formula> <tex-math notation="LaTeX">$\mathcal {AZI}$ </tex-math></inline-formula> over <inline-formula> <tex-math notation="LaTeX">$\Omega \left ({n,p}\right) $ </tex-math></inline-formula>. We first show that under a certain condition, the problem reduces to finding the maximal value of <inline-formula> <tex-math notation="LaTeX">$\mathcal {AZI}$ </tex-math></inline-formula> over <inline-formula> <tex-math notation="LaTeX">$\Omega _{1}\left ({n,p}\right) $ </tex-math></inline-formula>, the set of trees in <inline-formula> <tex-math notation="LaTeX">$\Omega \left ({n,p}\right) $ </tex-math></inline-formula> with no vertices of degree 2. Then we rely on this result to find the trees with maximal value of <inline-formula> <tex-math notation="LaTeX">$\mathcal {AZI}$ </tex-math></inline-formula> over <inline-formula> <tex-math notation="LaTeX">$\Omega \left ({n,p}\right) $ </tex-math></inline-formula>, when <inline-formula> <tex-math notation="LaTeX">$p=2$ </tex-math></inline-formula> and 3. In particular, we deduce that the conjecture holds for all trees with at most 3 branching vertices.

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