The neighborhood version of the hyper Zagreb index of trees

Topological indices are numeric quantities that describes the topology of molecular structure in mathematical chemistry. One of these indices is the neighborhood version of the hyper Zagreb index, [Formula: see text]. The neighborhood version of the hyper Zagreb index is expressed as the sum over all pairs of adjacent vertices [Formula: see text] and [Formula: see text] of the terms [Formula: see text], where [Formula: see text] is the neighborhood degree sum of the vertex [Formula: see text] which is the sum of degrees of all of its neighboring vertices. In this paper, we show that over all simple connected graphs, the path graph [Formula: see text] has the minimum value of [Formula: see text] and the complete graph [Formula: see text] has the maximum value of [Formula: see text]. Also, we give the minimum value of this index within the set of trees with given vertices number and maximum vertex degree and specify the minimal trees.