On the maximum ABC index of bipartite graphs without pendent vertices

Abstract For a simple graph G, the atom–bond connectivity index (ABC) of G is defined as ABC(G) = ∑ u v ∈ E ( G ) d ( u ) + d ( v ) − 2 d ( u ) d ( v ) , $\sum_{uv\in{}E(G)} \sqrt{\frac{d(u)+d(v)-2}{d(u)d(v)}},$where d(v) denotes the degree of vertex v of G. In this paper, we prove that for any bipartite graph G of order n ≥ 6, size 2n − 3 with δ(G) ≥ 2, A B C ( G ) ≤ 2 ( n − 6 ) + 2 3 ( n − 2 ) n − 3 + 2 , $ABC(G)\leq{}\sqrt{2}(n-6)+2\sqrt{\frac{3(n-2)}{n-3}}+2,$and we characterize all extreme bipartite graphs.


Introduction
Let G be a simple connected graph with vertex set V = V(G) and edge set E = E(G), in which a simple graph contains no loops and duplicate edges between two vertices. The order |V| of G is denoted by n = n (G), and the size |E| of G is denoted by m = m(G). For every vertex v ∈ V, the open neighborhood N(v) is the set {u ∈ V(G) | uv ∈ E(G)}. The degree of a vertex v ∈ V is d G (v) = d(v) = |N(v)|. The minimum degree of a graph G are denoted by δ = δ(G). A leaf or a pendant vertex of a graph G is a vertex of degree 1. For a subset S of vertices of G, we denote by G[S] the subgraph induced by S. A bipartite graph is a graph having no odd cycles.
The connectivity index, χ, is a topological index introduced in 1975 by Milan Randić [1] who has shown this index to reflect molecular branching. However, many physico-chemical properties are dependent on factors rather different than branching. In order to take this into account along with keeping the spirit of the Randić in-dex, Ernesto Estrada et al. [2] proposed a new topological index, named the atom-bond connectivity (ABC) index. Estrada [3] proved that the ABC index provides a good model for the stability of linear and branched alkanes as well as the strain energy of cycloalkanes. For those who may not be familiar with alkanes, these are any of the series of saturated hydrocarbons having the general formula C n H 2n+2 , including methane, ethane and propane. Additional physico-chemical applicabilities of the ABC index were presented in a few other works, including [3][4][5]. The atom-bond connectivity index of G is defined as Due to its physico-chemical applicability, the ABC index has attracted significant attention from researchers in recent years, and many mathematical properties of this index have been reported [5][6][7][8][9][10][11][12][13][14][15].
When examining a topological index, one of the fundamental questions that needs to be answered is for which graphs this index assumes minimal and maximal values. In addition, the identities of these extremal values must be uncovered. The minimal value of the ABC index has been investigated by several authors [16][17][18][19][20][21][22], and the maximal value of the ABC index has been studied extensively, as well [23][24][25][26]. Recently, Shao et al. [15,27] showed that for any connected graph G of order n, size 2n − 4 ≥ 4 with δ(G) ≥ 2, ABC(G) ≤ (n − 2) √ 2 characterizes all extremal graphs. They also proved that for any connected graph G of order n, size 2n − 3 ≥ 5 with δ(G) ≥ 2, characterizes all extremal (n, 2n − 2) graphs without pendent vertices and with maximum ABC indices.
We make use of the following results and notations in this paper. To simplify notations, we define the following functions: Similarly, we can see that h y(x, y) > 0. Let K 3,t be the complete bipartite graph with bipartite sets X = {x 1 , x 2 , x 3 } and Y = {y 1 , . . . , y t }. For s ≥ 6, assume H 3,t s is the bipartite graph obtained from K 3,t by adding s vertices v 1 , . . . , vs and joining them to x 1 and x 3 (see Figure 1). If s = 0, we define H 3,t s to be the graph K 3,t . The proof of next result is easy to verify by direct calculation.
In the rest of paper, we employ the following notation defined in [15]. and In the following, we will omit the superscript where no confusion can arise. (v) . When no confusion can arise, we simplify ABC(e|G) to ABC(e).

Main results
In this section, we present an upper bound on the ABC index of a bipartite graph and characterize all extreme bipartite graphs. For any bipartite graph G, we let (X G , Y G ) denote its bipartition. Proof: We first perform the following steps: Step 1: Using the software geng in package nauty [29], we generate the set of all graphs of order n ∈ {6, 7, 8, 9}, size 2n − 3 with δ(G) ≥ 2.
Step 2: For each graph G in the obtained graphs, we compute ABC(G) according to formula (1). It turns out that the result holds for each n ∈ {6, 7, 8, 9}. Now, assume that n ≥ 10. Let G n be the family of graphs G of order n, size 2n − 3, minimum degree δ(G) ≥ 2, the maximum ABC index and different from H 3,3 n−6 . We will show that G n = ∅. Suppose, to the contrary, that Gn ≠ ∅ for some n ≥ 10. We further assume that n is as small as possible such that G n ≠ ∅. Let G ∈ Gn. We proceed with establishing several claims. Claim 1. k ≥ 6, |E 2,2 | = 0 and 2k − 3 = |E 3 + ,3 + |.

Case 1. u ∉ N(s) and v ∉ N(t).
Let G be the graph obtained from G by removing two edges uv, st and adding the edges us, tv. Clearly, G is a bipartite graph of order n, size 2n − 3 with δ(G ) ≥ 2. By Lemma 2, we have which is a contradiction.
We distinguish two subcases.
and v ∉ N(p). As in Case 1, we obtain a contradiction again.
Proof of Claim 2. Suppose, to the contrary, that E 2,3 ≠ ∅ and let uv ∈ E 2,3 . Assume without loss of generality that d(v) = 2 and d(u) = 3. Since E 2,2 = ∅, the other neighbor of v, say t, is in V 3 + . We distinguish the following cases.
we obtain which is a contradiction.
Suppose G is the graph obtained from G by removing the edges vu, sh and adding the edges vs and uh. Clearly, G is bipartite of order n, size 2n − 3 and δ(G ) ≥ 2. Since ABC(uh|G ) ≥ ABC(sh|G) and Considering the edge uw ∈ G , we are in situation (a) and we obtain a contradiction. Now let s ∈ N(v). Using an argument similar to that described in Case 2 of Claim 1, we deduce that there exists an edge pq ∈ E 3 + ,3 + such that s ∉ {p, q}. Using the proof of Case (b), we obtain a contradiction again.
This completes the proof of Claim 2.

Claim 3.
There exists a bipartite graph G = (X G , Y G ) of order n, size 2n − 3 and δ(G ) ≥ 2, which satisfies the following conditions: Proof of Claim 3. For any bipartite graph G 1 of order n, size 2n − 3 and δ(G 1 ) ≥ 2, we define and L( is as small as possible. Clearly, G satisfies in the condition (i). Then G ∈ Gn and so G satisfies Claims 1 and 2. If G satisfies in the conditions (ii) and (iii) of Claim 3, then we are done. Assume that G does not satisfy in the condition (ii) or (iii). Assume without loss of generality that G does not satisfy condition (ii). Then Then there are two vertices u 1 , We consider the following cases.
Let G s be the graph obtained from G by removing the edges v 1 a i (1 ≤ i ≤ k 1 ) and adding the edges v 3 a i (1 ≤ i ≤ k 1 ). Clearly, Gs is a bipartite graph of order n, size 2n − 3 with δ(G s) ≥ 2. and Assume that l = ABC(Gs) − ABC(G ). We have We conclude from Lemma 1 and the fact d(v 3 |G )−d(v 1 |G )+ k 1 ≥ 1 that l = ABC(Gs) − ABC(G ) > 0, which leads to a contradiction as above.
We deduce from Lemma 1 that ABC(Gs) − ABC(G ) > 0. This implies that ABC(G s) > ABC(G ), which is a contradiction. 2 for any edge e ∈ E G v4 . Hence ABC(G s) ≥ ABC(G ) and L(Gs) ≤ L(G ). We deduce from the choice of G that L(Gs) = L(G ). Now, we consider Gs instead of G and proceed as Case A to obtain a contradiction.
It follows from Lemma 1 and the fact d which is a contradiction. Subcase B.4. 0 < q 2 < q 1 . Let G s be the graph obtained from G by removing the edges v 2 a i , v 4 b i (1 ≤ i ≤ q 2 ) and adding the edges . Clearly, Gs is a bipartite graph of order n, size 2n − 3 with δ(G s) ≥ 2 and Gs ≠ H 3,3 n−6 . It is easy to verify that ABC(G s) ≥ ABC(G ) and L(Gs) ≤ L(G ). Considering G s instead of G and applying the argument similar to that described in Subcases A.2 and A.3, we obtain a contradiction. This completes the proof of Claim 3.

ABC(G ) +
which is a contradiction.
and ABC (G ) = uv∈E(G ) ABC (uv). By Claims 1 and 2,  3 + 2 (these two graphs G 1 9 and G 2 9 illustrated in Figure 2). It is easy to verify that d(u) + d(v) ≥ 6 for every two vertices u, v ∈ V(G 1 9 ) or u, v ∈ V(G 2 9 ), a contradiction with Claim 4. Now let k = 8. Then ABC(G) ≤ (n − 8) illustrated in Figure 2). If G[V 3 + ] = G 1 8 , then it is easy to verify d(u) + d(v) ≥ 6 for every two vertices u, v ∈ V(G 1 8 ), a contradiction with Claim 4. If G[V 3 + ] = G 2 8 , then let u be the vertex of degree 2, and assume without loss of generality that u ∈ X. Suppose v ∈ X be a vertex of degree 3. By Claim 4, we conclude that G is a graph obtained from G 2 8 by adding vertices a 1 , a 2 , . . . , a n−8 and the edges a i u, a i v for each i ∈ {1, 2, . . . , n − 8} (See graph F 8 in Figure 3). It is not hard to verify that ABC(G) < ABC(H 3,3 n−6 ), a contradiction. Finally let k = 7. A computer search shows that there are exactly two bipartite graphs G satisfying the conditions |V(G )| = 7, |E(G )| = 11. We deduce from Claim 4 that G is a graph obtained from G by adding new vertices v 1 , v 2 , . . . , v n−7 and joining then to two vertices u, v ∈ X G (see the graph F 7 illustrated in Figure 3). It is easy to verify that ABC(F 7 ) < ABC(H 3,3 n−6 ), a contradiction. This completes the proof of Claim 6.  By applying the approach used above, we also obtain the following result:

Discussions
We studied the atom-bond connectivity (ABC) index on bipartite graphs, in which the ABC index has an important application in rationalizing the stability of linear and branched alkanes as well as the strain energy of cycloalkanes. All extremal graphs attained at these extremal values are characterized. Our method is to use monotonic functions and combine them with graph operations. Some special graphs and their values are obtained by computational searches. Our results extend the previous outcomes and deduce all bounds.