The minimal augmented Zagreb index of k-apex trees for k∈{1,2,3}

For a graph G containing no component isomorphic to the 2-vertex path graph, the augmented Zagreb index (AZI) of G is defined as AZI(G)=∑uv∈E(G)([Formula presented])3.This topological index has been proved to be closely correlated with the formation heat of heptanes and octanes. A k-apex tree is a connected graph G admitting a k-subset of vertices X such that G−X is a tree, but for any subset of vertices X′ of order less than k, G−X′ is not a tree. In this paper, we determine the minimum AZI among all k-apex trees for k∈{1,2,3}.