Abstract
Let G 1 ∘ 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 V G 1 copies of G 2 and then joining the i th vertex of G 1 to every vertex in the i th copy of G 2 . The atom-bond connectivity index (ABC index) of a graph G is defined as A B C G = ∑ u v ∈ E G d G u + d G v − 2 / d G u d G v , where E G is the edge set of G and d G u and d G v are degrees of vertices u and v , respectively. For the ABC indices of G 1 ∘ 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 ∘ 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 ∘ 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 ∘ T 2 attains the maximum value if and only if T 1 is a path and T 2 is a star.
Highlights
Graph theory has been applied in many engineering fields such as mechanical design and manufacturing and chemical engineering
The link in the mechanism can be regarded as the vertex, kinematic pair can be regarded as an edge, and the topological configuration of the mechanism is abstracted as the graph. erefore, the nature and characteristics of the mechanism can be analyzed by relevant graph theory
If a chemical molecule is regarded as a two-dimensional graph, the graph’s vertices represent atoms, and edges represent chemical bonds, the graph determines the topological properties of the given molecule
Summary
Graph theory has been applied in many engineering fields such as mechanical design and manufacturing and chemical engineering. It is evident that Kn has the maximal ABC index, whereas the connected graph with the minimal ABC index must be a tree (see [2, 3]). In [13], the lower and upper bounds for ABC indices of edge corona product of graphs were given. In [14], the extremal edge-version ABC index of some graph operations was given. Lemmas 5 and 6 show that deleting an edge in one of the graphs (G1 and G2) will decrease the ABC index of G1 ∘ G2. Note that both trees T2 and T3 have the same vertex set, and except u1 and u2, the other vertices have the same edges in T2 and T3. Let T1 and T2 be two trees of orders n1 ≥ 3 and n2 ≥ 3, respectively. en, ABC T1 ∘ T2 ≤ ABCT1 ∘ Sn2,
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