Independent Neighborhood Polynomial of the Direct and Corona Products of Trees

Let G be a connected graph. We say that a given graph is a tree if every pair of vertices is connected by a unique path. The corona product of two graphs G and H is defined as the graph obtained by taking one copy of G and |V (G)| copies of H and joining the ith vertex of G to every vertex in the ith copy of H. On the other hand, the direct product G × H is a graph such that the vertex set of G× H is the Cartesian product V (G)×V (H); and vertices (g, h) and (g′, h′) are adjacent in G×H if and only if g is adjacent to g′ in G, and h is adjacent to h′in H.