Maximal Augmented Zagreb index of trees with at most three branching vertices (2019)
The Augmented Zagreb index of a graph G is defined to be AZI(G)=∑uv∈E(G)(d(u)d(v)d(u)+d(v)−2)3 , where E(G) is the edge set of G , d(u) and d(v) 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 AZI among trees. However, the problem of finding the tree with maximal 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 AZI among all trees with n vertices, for all n≥19 . Let Ω(n,p) be the set of trees with n vertices and p branching vertices. In this paper we consider the maximal value problem of AZI over Ω(n,p) . We first show that under a certain condition, the problem reduces to finding the maximal value of AZI over Ω1(n,p) , the set of trees in Ω(n,p) with no vertices of degree 2. Then we rely on this result to find the trees with maximal value of AZI over Ω(n,p) , when p=2 and 3. In particular, we deduce that the conjecture holds for all trees with at most 3 branching vertices.