The Maximal ABC Index of the Corona of Two Graphs

Let G(1)omicron G(2) be the corona of two graphs G(1) and G(2) which is the graph obtained by taking one copy of G(1) and VG(1) copies of G(2) and then joining the ith vertex of G(1) to every vertex in the ith copy of G(2). The atom-bond connectivity index (ABC index) of a graph G is defined as ABCG= Sigma(uv is an element of E(G))root(d(G)u + d(G) (v) -2/d(G) (u)d(G) (v), where EG is the edge set of G and dGu and dGv are degrees of vertices u and v, respectively. For the ABC indices of G(1) omicron G(2) with G(1) and G(2) being connected graphs, we get the following results. (1) Let G(1) and G(2) be connected graphs. The ABC index of G(1) o G(2) attains the maximum value if and only if both G(1) and G(2) are complete graphs. If the ABC index of G(1) o G(2) attains the minimum value, then G(1) and G(2) must be trees. (2) Let T-1 and T-2 be trees. Then, the ABC index of T-1 o T-2 attains the maximum value if and only if (T)1 is a path and T-2 is a star.