The Maximal ABC Index of the Corona of Two Graphs

Let G1 ∘G2 be the corona of two graphs G1 and G2 which is the graph obtained by taking one copy of G1 and |V(G1)| copies of G2 and then joining the ith vertex of G1 to every vertex in the i th copy of G2. *e atom-bond connectivity index (ABC index) of a graph G is defined as ABC(G) � 􏽐uv∈E(G) ��������������������������� (dG(u) + dG(v) − 2/dG(u)dG(v)) 􏽰 , where E(G) is the edge set of G and dG(u) and dG(v) are degrees of vertices u and v, respectively. For the ABC indices of G1 ∘G2 with G1 and G2 being connected graphs, we get the following results. (1) Let G1 and G2 be connected graphs. *e ABC index of G1 ∘G2 attains the maximum value if and only if both G1 and G2 are complete graphs. If the ABC index of G1 ∘G2 attains the minimum value, then G1 and G2 must be trees. (2) Let T1 and T2 be trees. *en, the ABC index of T1 ∘T2 attains the maximum value if and only if T1 is a path and T2 is a star.


Introduction
Graph theory has been applied in many engineering fields such as mechanical design and manufacturing and chemical engineering. In the graph theory, the link in the mechanism can be regarded as the vertex, kinematic pair can be regarded as an edge, and then 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. In chemical engineering, if a chemical molecule is regarded as a two-dimensional graph, the graph's vertices represent atoms, and edges represent chemical bonds, then the graph determines the topological properties of the given molecule.
Molecular descriptors play an important role in chemistry and pharmacology. Among these molecular descriptors, so-called topological indices play a significant role. Topological indices are the mathematical tools that correlate the chemical structure with various physical properties, chemical reactivity, or biological activity numerically. And the topological indices have been widely applied in the study of the stability of alkanes and the strain energy of cycloalkanes. In the field of pharmaceutical chemistry and bioinformatics, topological index can be used to encode the chemical structure. is encode strategy provides the annotation, comparison, rapid collection, mining, and retrieval of chemical structures within large databases. Afterward, topological indices can be used to look for quantitative structure-activity relationships and quantitative structureproperty relationships. In QSAR/QSPR studies, the biological activities of compounds can be predicted according to their topological indices, such as Zagreb, Randic, and the atom-bond connectivity indices. e topological indices can be classified by the structural properties of graphs used for their calculation. e atombond connectivity index, which was proposed by Estrada et al. in 1998, is a vertex-degree-based graph topological index.
We consider finite undirected connected graphs without loops or multiple edges. Let G 1 and G 2 be two such graphs. e corona of G 1 and G 2 , denoted by G 1 ∘ G 2 , is defined as the graph obtained by taking one copy of G 1 and |V(G 1 )| copies of G 2 and then joining the ith vertex of G 1 to every vertex in the ith copy of G 2 (as shown in Figure 1). Corona graphs were introduced by Frucht and Harary in 1970 [1].
where d G (u) (or d u ) and d G (v) (or d v ) are degrees of vertices u and v, respectively. Let G be a graph. For an edge e � uv ∈ E(G) and an edge set We say that ABC G (e) is the ABC index of e and ABC G (E 0 ) is the ABC index of E 0 .
roughout the paper, we use K n , S n , and P n to denote the complete graph, the star, and the path of order n, respectively. e ABC index attracted a lot of attention in the last few years. Several properties of ABC index were established. In particular, if a new edge is inserted into G, then ABC index necessarily increases (see Lemma 1).
Lemma 1 (see [2]). Let G be a simple graph with nonadjacent vertices i and j. en, It is evident that K n has the maximal ABC index, whereas the connected graph with the minimal ABC index must be a tree (see [2,3]). e smallest ABC index of trees with n pendant vertices was characterized in [4]. In contrast to the minimal case, the tree with the maximal ABC index was easily identified as the star (see [5]). In [6], the maximum and minimum ABC indices of all unicyclic graphs and unicyclic chemical graphs were obtained, and the corresponding extremal graphs were also characterized. In [7,8], the maximum values of the ABC indices in the class of all n-vertex bicyclic and tricyclic graphs were presented, respectively.
Recently, some researchers have paid more attention to the ABC spectral radius which is associated with ABC energy.
Chen [9] characterized the graphs with extremal ABC spectral radius for a class of given graphs. Lin et al. [10] determined the trees with the third, fourth, and fifth largest ABC spectral radii.
For corona graph, Bian et al. [11] considered some Wienertype indices of the corona graphs. Lu and Xue [12] studied the Kirchhoff index of two corona graphs. 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. For more information, see [15][16][17][18][19].
Motivated by this, it is interesting to determine the extremal graphs among the set G 1 ∘ G 2 |G 1 and G 2 are connected graphs (trees)}. From Lemma 1, the problem is simple if G 1 and G 2 are connected graphs, so we give some conclusions directly. We mainly consider the problem for the case that G 1 and G 2 are trees. e main results are as follows.
(1) Let G 1 and G 2 be connected graphs. e 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. e 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.
. en, f(x, y) strictly decreases with x for fixed y ≥ 2, and the function zf/zx increases with x. Similarly, the function f(x, y) strictly decreases with y for fixed x ≥ 2, and the function zf/zy increases with y.

The Bounds of ABC Indices of G 1 ∘ G 2
Let G 1 and G 2 be two simple connected graphs of orders n 1 and n 2 , respectively, and en, the following four conditions hold: From (C4) and (C3), the following theorem is obtained immediately, and we omit the proof. Theorem 1. Let G 1 and G 2 be two simple connected graphs of orders n 1 and n 2 , respectively, and G � G 1 ∘ G 2 . en, By Lemma 1 and eorem 1, the following results are clear.

Lemma 5.
Let G 1 and G 2 be two simple connected graphs of orders n 1 ≥ 3 and n 2 ≥ 3, respectively. en, for any edge Lemma 6. Let G 1 and G 2 be two simple connected graphs of orders n 1 ≥ 3 and n 2 ≥ 3, respectively. en, for any edge Theorem 2. Let G 1 and G 2 be two simple connected graphs of orders n 1 ≥ 3 and n 2 ≥ 3, respectively. en, and the equality holds if and only if G 1 � K n 1 and G 2 � K n 2 .
Lemmas 5 and 6 show that deleting an edge in one of the graphs (G 1 and G 2 ) will decrease the ABC index of G 1 ∘ G 2 . Consequently, we get the necessary condition for the ABC indices of G 1 ∘ G 2 to be the minimum.

Theorem 3.
If the ABC index of G 1 ∘ G 2 attains the minimum value, then G 1 and G 2 must be trees.

The Upper Bound of ABC Indices of T 1°T2
Since the minimal ABC trees are not unique (see [2,5]), it seems to be difficult to characterize the corona of two trees with minimal ABC index. We leave the problem as a future task. In this section, we are going to give the upper bound of ABC indices for T 1 ∘ T 2 . Lemma 7. Let T 1 be any tree of order n 1 ≥ 3 and T 2 be a tree of order n 2 ≥ 4 as depicted in Figure 2, where u 1i and u 2j are pendant vertices for i � 1, . . . , s and j � 1, . . . , t with s ≥ 1, Proof. Denote T � T 1 ∘ T 2 , and T ′ � T 1 ∘ T 3 . Let v be any vertex of T 1 . We can get the result if we prove that Note that both trees T 2 and T 3 have the same vertex set, and except u 1 and u 2 , the other vertices have the same edges in T 2 and T 3 . So, Mathematical Problems in Engineering 3 en, by Lemma 4 (taking x � d T 2 (u 2 ) + 1, y � d T 1 (v) + n 2 , and z � d T 2 (u 1 ) + 1), e lemma holds. Using Lemma 7 repeatedly, we can obtain the following result.
□ Corollary 1. Let T 1 and T 2 be two trees of orders n 1 ≥ 3 and n 2 ≥ 3, respectively. en, and the equality holds if and only if T 2 � S n 2 .
Theorem 4. Let T 1 and T 2 be two trees of orders n 1 ≥ 3 and n 2 ≥ 3, respectively. en, and the equality holds if and only if T 2 � S n 2 .

Note that
By Lemma 3, and equality holds if and only if d T 2 (u) + 1 � 2 or d T 2 (v) + 1 � 2 for each edge uv ∈ E(T 2 ), that is, T 2 � S n 2 . e result follows now. □ Theorem 5. Let T 1 and T 1 ′ be two trees as depicted in Figure 3, which contain T 0 as a subtree, where

ABC T
Proof. Denote T � T 1 ∘ S n 2 and T ′ � T 1 ′ ∘ S n 2 . Note that ABC T (E 3 (T)) � ABC T′ (E 3 (T ′ )). We can get the result if we show Firstly, we consider ABC T (E 2 (T)) − ABC T′ (E 2 (T ′ )). Let u be any vertex of T 1 (T 1 ′ ). If w ∈ V(S n 2 ) is a pendant vertex of S n 2 , then the ABC index of uw is ��� 1/2 √ . If u is a pendant vertex of T 1 and T 1 ′ and w ∈ V(S n 2 ) is the central vertex of S n 2 , then the ABC index of uw has no change. So, we only need to calculate the ABC index of uw, where u is not a pendant vertex of T 1 ′ and w is the central vertex of S n 2 . Denote Combining the situation of ABC T (E 1 (T)) − ABC T′ (E 1 (T ′ )), we consider the following three cases.
By Lemma 2, 1 (a, b).  1 (a, b) has the following properties : (1) e function f 1 (a, b) is a strictly decreasing function on b for fixed a ≥ 6.
We can see the result from the graph of f 1 (a, b). For example, the graphs of f 1 (6, b) and f 1 (10, b) are as follows (as depicted in Figures 4 and 5). So, (2) e maximum value of the function f 1 (a, a + 1) is attained when a is sufficiently large.
We can get this result from the graph (as depicted in Figure 6) of f 1 (a, a + 1). en, So, for b > a ≥ 6, f 1 (a, b) < 0 and the theorem follows for Case 1.
By Lemma 2, Similar to the discussions in Case 1, we have that the function f 2 (a, b) is a strictly decreasing function on b for fixed a ≥ 6, and the maximum value of the function f 2 (a, a + 1) is attained when a is sufficiently large. So, e theorem holds for Case 2.
e values of ABC indices of T 1°T2 for n 1 < 3, n 2 < 4 are as follows.

Conclusions
In this paper, we mainly determined the upper bounds of the ABC indices of T 1 ∘ T 2 . It is evident that 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°G2 attains the minimum value, then G 1 and G 2 must be trees. We deduced that 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. We will discuss the minimum value of the ABC indices of the corona of two trees in near future.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.