The minimal-ABC trees with B2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_2$$\end{document}-branches

The atom–bond connectivity (ABC) index is a degree-based graph topological index with a lot of chemical applications, including those of predicting the stability of alkanes and the strain energy of cycloalkanes. It is known (Chen and Guo in MATCH Commun Math Comput Chem 65:713–722, 2011; Das et al. in MATCH Commun Math Comput Chem 76:159–170, 2011) that among all connected graphs, trees minimize the ABC index (such trees are called minimal-ABC trees). Several structural properties of minimal-ABC trees were proved in the past several years. Here we continue to make a step forward towards the complete characterization of the minimal-ABC trees. In Dimitrov (Discrete Appl Math 204:90–116, 2016), it was shown that a minimal-ABC tree cannot contain more than 11 so-called B2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_2$$\end{document}-branches. We improve this result by showing that if a minimal-ABC tree of order larger than 39 contains so-called B1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_1$$\end{document}-branches, then it contains exactly one B2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_2$$\end{document}-branch, and if a minimal-ABC tree of order larger than 163 contains no B1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_1$$\end{document}-branch, then it contains at most two B2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_2$$\end{document}-branches.


Introduction and preliminaries
The atom-bond connectivity (ABC) index is a vertex-degree-based graph invariant, which attracted a lot of attention in the past few years. It was proposed in 1998 by Estrada et al. (1998), who showed that it can be a valuable predictive tool in the study of the heat of Communicated by Maria Aguieiras de Freitas.
B Darko Dimitrov darko.dimitrov11@gmail.com Zhibin Du zhibindu@126.com formation in alikeness. Ten years later, Estrada (2008) elaborated a novel quantum-theorylike justification for this topological index. After that revelation, the interest of ABC-index has grown rapidly. However, several problems persist open, though. For example, it is known that the star of a given order has the maximal ABC index among trees Furtula et al. (2009). Although a number of important properties of the trees with minimal ABC index are known, a full characterization of them is still an open problem. For some conjectures and partial results, the reader can be referred to , Gutman et al. (2013), Lin et al. (2013), Liu and Chen (2014). More theoretical and computational results regarding the ABC index of graphs can be, for example, found in Das et al. (2016), Dimitrov et al. (2016); Dimitrov (2017), Dimitrov and Milosavljević (2018), Du and da Fonseca (2016), Gao and Shao (2016), Lin et al. (2016Lin et al. ( , 2018. For a given simple graph G = (V , E), let us denote by d u the degree of vertex u, and uv the edge incident to the vertices u and v. Then the ABC index is defined as In the sequel, we mention some definitions and results, which are crucial in the further investigation of the minimal-ABC trees.
In Wang (2008), Wang defined the greedy trees, for a given degree sequence, as follows: Definition 1.1 Suppose that the degrees of the non-leaf vertices are given, the greedy tree is achieved by the following 'greedy algorithm': Fig. 1 The B k -, B * k , and B * * 3 -branches, for k ≥ 1 Theorem 5 (Dimitrov 2016) A minimal-ABC tree contains at most four B 1 -branches.
Recently, the following improvements of the above results were obtained: Theorem 6 (Dimitrov et al. 2018b) A minimal-ABC tree cannot contain four B 1 -branches.
Theorem 7 (Dimitrov et al. 2018b) A minimal-ABC tree cannot contain three B 1 -branches.
Lemma 8 ) If a minimal-ABC tree of order larger than 122 contains B 1 -branches, then the number of B 1 -branches can only be two, and such two B 1 -branches belong to a B * * 3 -branch.

Lemma 9 (Dimitrov 2016) Let w be a vertex of a minimal-ABC tree T which is not the root of T . Then w is a parent of at most six B 2 -branches.
Lemma 10 (Dimitrov 2016) A minimal-ABC tree does not contain more than 11 B 2 -branches.
The following two technical lemmas will be used in the proofs of several results presented in the next sections: Lemma 11 Dimitrov (2014) Let g(x, y) with real numbers x, y ≥ 2, x ≥ 0, 0 ≤ y < y. Then g (x, y)

increases in x and decreases in y.
Due to the symmetry of the function g(x, y), we can also get an equivalent version of Lemma 11.

decreases in x and increases in y.
In the rest of the paper, we consider the number of B 2 -branches that a minimal-ABC tree may contain. Our main achievement is a significant improvement of the existing upper bound on the number of B 2 -branches presented in Lemma 10. Namely, we show that if a minimal-ABC tree of order larger than 39 contains B 1 -branches, then it contains exactly one B 2 -branch, and if a minimal-ABC tree of order larger than 163 contains no B 1 -branch, then it contains at most two B 2 -branches.

Some non-existent local structures of minimal-ABC trees
In this section, we will show the non-existence of some local structures related to B 2 -branches in minimal-ABC trees.
First we consider the cases when there are 3, 4, 5, 6 B 2 -branches sharing a common parent in minimal-ABC trees.
Lemma 13 When t = 3, 4, 5, 6, then the configuration T depicted in Fig. 2 cannot occur in a minimal-ABC tree.
Proof We partition our proof into four cases according to the value of t. Let d x be the degree of vertex x in T .
For s ≥ 25, we apply the transformation T 1 illustrated in Fig. 3. After applying T 1 , the change of the ABC index is (1) Let g(x) = x · f (x + 1, 4) for x > 1. It can be easily verified that g (x) > 1 2 for x > 1. Then we have Note that 1 s+3 − 1 s+4 decreases in s > 0. And by Lemma 12, we have f (s + 4, 4) − f (s + 4, 3) decreases in s > 0. It follows that the right-hand side of (3) also decreases in s > 0.  For s ≥ 25, we have For 0 ≤ s ≤ 24, we apply the transformation T 2 illustrated in Fig. 4. Let E(T ) be the set of edges inT . After applying T 2 , the change of the ABC index is When s = 0, notice that Clearly, the right-hand side of (6) decreases in d v ≥ 4. For d v ≥ 8, we have On the other hand, it is easily verified that (s + 1) f (d v + s + 1, 4) − f (d v , s + 4) decreases in d v ≥ s + 4 when 1 ≤ s ≤ 24, which implies that the right-hand side of (5) also decreases in d v ≥ s + 4 (together with the fact that f (d v + s + 1, 5) decreases in d v ≥ s + 4). Now by direct calculation, we may deduce that ABC(T 1 ) < ABC(T ) holds for the following cases: Besides the upper bound about ABC(T 1 ) − ABC(T ) as (5), by considering in (4) the term we may get a somewhat stricter upper bound about ABC(T 1 ) − ABC(T ).
Note that, from Lemma 11, and recall that every neighbor of v inT has degree at least 3 when s = 0, and at least 4 when s ≥ 1; thus when s ≥ 1. Now it follows that when s = 0, and For the remaining cases: we would turn to use (7) (when s = 0) and (8) (when s ≥ 1), and a negative upper bound, equivalently ABC(T 1 ) < ABC(T ), follows from direct calculation easily.
Case 2. t = 4. For s ≥ 23, we apply the transformation T 3 illustrated in Fig. 5. After applying T 3 , the change of the ABC index is By Lemma 11, we get that and thus When 23 ≤ s ≤ 29, the right-hand side of (10) is negative, i.e., ABC(T 1 ) < ABC(T ), follows from direct calculation.
For s ≥ 30, we have For 0 ≤ s ≤ 22, we apply the transformation T 4 illustrated in Fig. 6. After applying T 4 , the change of the ABC index is It is easily verified that (s , which implies that the right-hand side of (13) also decreases in d v ≥ s + 5. Now by direct calculation, we may deduce that ABC(T 1 ) < ABC(T ) holds for the following cases: Fig. 6 The transformation T 4 in the proof of Lemma 13 when t = 4, 5 or 6 with small s Besides the upper bound on ABC(T 1 ) − ABC(T ) as (13), by considering in (12) the term we may get a somewhat stricter upper bound on ABC(T 1 ) − ABC(T ).
Note that, from Lemma 11, and recall that every neighbor of v inT has degree at least 3 when s = 0, and at least 4 when s ≥ 1; thus when s = 0, and when s ≥ 1. For the remaining cases: we would turn to use (14) (when s = 0) and (15) (when s ≥ 1), and a negative upper bound, equivalently ABC(T 1 ) < ABC(T ), follows from direct calculation easily.
Case 3. t = 5. For s ≥ 14, we apply the transformation T 5 illustrated in Fig. 7. After applying T 5 , the change of the ABC index is By Lemma 11, we get that and thus When 14 ≤ s ≤ 23, the right-hand side of (17) is negative, i.e., ABC(T 1 ) < ABC(T ), follows from direct calculation.
For s ≥ 24, we have For 0 ≤ s ≤ 13, we apply the transformation T 4 illustrated in Fig. 6. After applying T 4 , the change of the ABC index is Clearly It is easily verified that decreases in the following cases: which implies that the right-hand side of (20) also decreases in such cases (together with the fact that f (d v + s + 3, 3) decreases in d v ≥ s + 6). Now by direct calculation, we may deduce that ABC(T 1 ) < ABC(T ) holds for the following cases: we may get a somewhat stricter upper bound on ABC(T 1 ) − ABC(T ) as that one in (20).
Note that, from Lemma 11, decreases in d x , and recall that every neighbor of v inT has degree at least 3; thus For the remaining cases: we would turn to use (21), and a negative upper bound on ABC(T 1 ) − ABC(T ) follows from direct calculation easily.
Case 4. t = 6. For s ≥ 9, we apply the transformation T 6 illustrated in Fig. 8. After applying T 6 , the change of the ABC index is By Lemma 11, we get that and thus When 9 ≤ s ≤ 19, the right-hand side of (23) is negative, i.e., ABC(T 1 ) < ABC(T ), follows from direct calculation.
For s ≥ 20, we have For 0 ≤ s ≤ 8, we apply the transformation T 4 illustrated in Fig. 6. After applying T 4 , the change of the ABC index is Clearly It is easily verified that (s + 3) f (d v + s + 4, 4) − f (d v , s + 7) decreases in the following cases: • d v ≥ 12 when s = 0; • d v ≥ 11 when s = 1; • d v ≥ 10 when s = 2; • d v ≥ s + 7 when s = 3, 4, 5, 6, 7, 8, which implies that the right-hand side of (26) also decreases in such cases (together with the fact that f (d v + s + 4, 3) decreases in d v ≥ s + 7). Now by direct calculation, we may deduce that ABC(T 1 ) < ABC(T ) holds for the following cases: Besides the upper bound on ABC(T 1 ) − ABC(T ) as (26), by considering in (25) the term

we may get a somewhat stricter upper bound on ABC(T 1 ) − ABC(T ).
Note that, from Lemma 11, f (d v + s + 4, d x ) − f (d v , d x ) decreases in d x , and recall that every neighbor of v inT has degree at least 3; thus For the remaining cases: we would turn to use (27), and a negative upper bound, equivalently ABC(T 1 ) < ABC(T ), follows from direct calculation easily. Now the proof is completed.
In the subsequent two lemmas, we will give some partial results when the number of B 2 -branches is 1 or 2.
Lemma 14 When t = 2, and s ≤ 22 or s ≥ 49, then the configuration T depicted in Fig. 2 cannot occur in a minimal-ABC tree.
For s ≥ 50, we have ≈ −0.0000381188. The case when s = 0 has been settled in (Dimitrov 2016, Proposition 4.1). For 1 ≤ s ≤ 22, we apply the transformation T 8 illustrated in Fig. 10. After applying T 8 , the change of the ABC index is On the one hand, it is easily verified that Clearly, the right-hand side of (33) decreases in d v ≥ s + 3. Now by direct calculation, we may deduce that ABC(T 1 ) < ABC(T ) holds for d v ≥ 13 when s = 1, 2.
On the other hand, it is easily verified that (s − 1) · f (d v + s, 4) − f (d v , s + 3) decreases in the following cases: which implies that the right-hand side of (32) also decreases in such cases (together with the fact that f (d v + s, 5) decreases in d v ≥ s + 3). Now by direct calculation, we may deduce that ABC(T 1 ) < ABC(T ) holds for the following cases: Besides the upper bound about ABC(T 1 ) − ABC(T ) as (32), by considering in (31) the term we may get a somewhat stricter upper bound about ABC(T 1 ) − ABC(T ).
Note that, from Lemma 11, decreases in d x , and recall that every neighbor of v inT has degree at least 4 when s > 0; thus (d v , 4)).
The proof is completed.
Lemma 15 When t = 1, and s ≤ 17, then the configuration T depicted in Fig. 2 cannot occur in a minimal-ABC tree.
Proof Note that s ≥ 2. For s = 2, we apply the transformation T 9 illustrated in Fig. 11. After applying T 9 , the change of the ABC index is It is easily verified that 4 f (d v + 3, 3) − f (d v , 4) decreases in d v ≥ 5, which implies that the right-hand side of (36) also decreases in d v ≥ 5. Now for d v ≥ 5, For the remaining case d v = 4, by direct calculation, one gets that the right-hand side of (36) is −0.0137609. For 3 ≤ s ≤ 17, we apply the transformation T 10 illustrated in Fig. 12. After applying T 10 , the change of the ABC index is On the one hand, it is easily verified that Clearly, the right-hand side of (39) decreases in d v ≥ s + 2. Now by direct calculation, we may deduce that ABC(T 1 ) < ABC(T ) holds for d v ≥ 18 when s = 3 and d v ≥ 20 when s = 4. On the other hand, it is easily verified that decreases in the following cases: which implies that the right-hand side of (38) also decreases in such cases (together with the fact that f (d v + s − 1, 5) decreases in d v ≥ s + 2). Now by direct calculation, we may deduce that ABC(T 1 ) < ABC(T ) holds for the following cases: Besides the upper bound about ABC(T 1 ) − ABC(T ) as (38), by considering in (37) the term we may get a somewhat stricter upper bound about ABC(T 1 ) − ABC(T ).
Note that, from Lemma 11, decreases in d x , and recall that every neighbor of v inT has degree at least 4 when s > 0, thus (d v , 4)).

Now it follows that
For the remaining cases: we would turn to use (40), and a negative upper bound, equivalently ABC(T 1 ) < ABC(T ), follows from direct calculation easily. Now the proof is completed.
The following two results will be used to show that a B 2 -branch and a B * * 3 -branch can not occur simultaneously.
Lemma 16 When s ≥ 11, the configuration T depicted as the left tree in Fig. 13, where possibly v = x, cannot occur in a minimal-ABC tree.
Proof Whether v = x or not, we always apply the transformation T 11 illustrated in Fig. 13. After applying T 11 , the change of the ABC index is By Lemma 12, f (s + 2, 4) − f (s + 2, 3) decreases in s > 0, which implies that the right-hand side of (41) also decreases in s > 0. For s ≥ 11, we have The proof is completed. Proof Whether v = x or not, we always apply the transformation T 12 illustrated in Fig. 14. After applying T 12 , the change of the ABC index is By Now by direct calculation, we get that the right-hand side of (43) is negative, i.e., ABC(T 1 ) < ABC(T ), for 23 ≤ s ≤ 48. The proof is completed.

New properties about B 2 -branches in minimal-ABC trees
Now we are ready to present some new structural properties related to B 2 -branches in minimal-ABC trees. A Kragujevac tree is a tree comprising of a single central vertex, B k -branches, with k ≥ 1, and at most one B * k -branch Hosseini et al. (2014).
Theorem 18 A minimal-ABC tree cannot contain a vertex having only B 2 -branches as its children.
Proof Let T be a minimal-ABC tree. Suppose to the contrary that u is a vertex of T which has only B 2 -branches as its children.
First assume that u is the root vertex of T . Note that now T is a Kragujevac tree. However, a Kragujevac tree with minimal ABC index would not be of such structure Hosseini et al. (2014).
Next assume that u is not the root vertex of T . Denote by v the parent vertex of u in T , i.e., T is of the form as depicted in Fig. 2 with s = 0. From Lemma 9, the number of B 2 -branches attached to u can only be 2, 3, 4, 5 or 6. Then the result follows from Lemmas 13 and 14. Now the proof is completed.
Theorem 19 A minimal-ABC tree contains at most two vertices having B 2 -branches as their children.
Proof Let T be a minimal-ABC tree. Suppose to the contrary that there are three vertices, say u, v, w, in T having B 2 -branches as their children. In the breadth-first search of T , assume that u occurs before v, and v occurs before w. By Proposition 1, v has only B 2 -branches as its children, which is a contradiction to Theorem 18.
Theorem 20 A minimal-ABC tree does not contain a B 2 -branch and a B * * 3 -branch that have a common parent vertex.
Proof Let T be a minimal-ABC tree. Suppose to the contrary that there is a B 2 -branch and a B * * 3 -branch in T sharing the same parent vertex, say u. Denote by w the child of u which is the root vertex of such B * * 3 -branch. Case 1. u is not the root vertex of T .
Denote by v the parent vertex of u in T . From Proposition 1, d v ≥ 4. Let z be a child of v in T different from u. If z occurs before u in the breadth-first search of T , then by Proposition 1, there are B 2 -branches attached to the children of z. If u occurs before z in the breadth-first search of T , then by Proposition 1, there are some B 2 -branches attached to z. In either case, we can find three vertices in T having B 2 -branches as their children, which is a contradiction to Theorem 19.
Case 2. u is the root vertex of T .
First we claim that w is the first child of u in the breadth-first search of T . Otherwise, there is a child of u, say y, occurring before w; then from Proposition 1, there are B 2 -branches attached to y, which means that there are three vertices (i.e., u, w, y) in T having B 2 -branches as their children, which is a contradiction to Theorem 19.
Again by Proposition 1, we get that every branch attached to u is a B 2 -or B 3 -branch, except the B * * 3 -branch, i.e., T is the tree as depicted on the left in Fig. 15. Now we apply the transformation T 13 illustrated in Fig. 15. After applying T 13 , the change of the ABC index is By Lemma 12, f (d u , 4) − f (d u , 3) decreases in d u ≥ 4; thus the right-hand side of (44) also decreases in d u ≥ 4. For d u ≥ 13, we have If 4 ≤ d u ≤ 12, then the order of T is at most 86; however, the minimal-ABC trees of such orders would not be of these structures Dimitrov and Milosavljević (2018). Finally the result follows.
Theorem 21 A minimal-ABC tree cannot contain a B 2 -branch and a B * * 3 -branch simultaneously.
Proof Let T be a minimal-ABC tree. Suppose to the contrary that there is a B 2 -branch and a B * * 3 -branch in T . Note that there are exactly two vertices, say u and w, in T having B 2 -branches as their children (from Theorem 19). Assume that u is the parent vertex of a B 2 -branch, and w is the root vertex of a B * * 3 -branch (it is also the parent vertex of another B 2 -branch). Clearly, u occurs before w in the breadth-first search of T .
From Theorem 20, we may assume that u is not the parent vertex of w. Note that u is not the root vertex of T . Denote by v the parent vertex of u in T .
First suppose that every branch attached to u is a B k -branch. From Theorem 2, Lemmas 3 and 8, k = 2 or 3, i.e., T is of the form as depicted in Fig. 2. Now by Lemmas 9 and 13, we get that u has at most two B 2 -branches as its children. Together with Lemmas 14, 15, 16, and 17, we know that a minimal-ABC tree cannot have such structure.
Otherwise, there is a child of u, say y, which has a child of degree more than 2. Then by Proposition 1, y has B 2 -branches as its children, i.e., there are three vertices (i.e., u, w, y) in T having B 2 -branches as their children, which is a contradiction to Theorem 19. Now the proof is completed.

The number of B 2 -branches in minimal-ABC trees
In the final section, our main results about the number of B 2 -branches in minimal-ABC trees are established.

Theorem 22 A minimal-ABC tree contains exactly one vertex having B 2 -branches as its children.
Proof Suppose to the contrary that u and w are the unique two vertices in T having B 2branches as their children. Assume that u occurs before w in the breadth-first search of T . By Proposition 1, the first child of w must be the root vertex of a B 2 -branch, and every branch attached to w is a B 2 -or B 1 -branch. Moreover, from Theorem 18, there must be a B 1 -branch attached to w, and from Lemma 8, w and all its descendants form a B * * 3 -branch. Finally, from Theorem 21, there is no other B 2 -branch in T , except the one in B * * 3 -branch, it means that w is the unique vertex in T having B 2 -branches as its children, which is a contradiction.
Theorem 23 Let T be a minimal-ABC tree. If T is of order larger than 39 and contains B 1 -branches, then T contains exactly one B 2 -branch. If T is of order larger than 163 and contains no B 1 -branch, then T contains at most two B 2 -branches.
Proof First suppose that T contains B 1 -branches. By Lemma 8, such B 1 -branches belong to a B * * 3 -branch. Note that a B * * 3 -branch contains exactly one B 2 -branch. Since T has exactly one vertex having B 2 -branches as its children, it implies that there is no other B 2 -branches in T , except the one in B * * 3 -branch, i.e., T contains exactly one B 2 -branch. Next suppose that T contains no B 1 -branch. Denote by u the unique vertex in T having B 2 -branches as its children.
Suppose that u is the root vertex of T . By Proposition 1, every grandchild of u in T is of degrees 1, 2 or 3. If all the grandchildren of u in T are of degrees 1 or 2, then all the branches attached to u are B k -branches, i.e., T is a Kragujevac tree. From Hosseini et al. (2014) and Dimitrov (2014), we know that a Kragujevac tree with minimal ABC index of order larger than 163 would contain at most two B 2 -branches. Otherwise, there is a grandchild of u in T of degree 3. Denote by y a child of u which has a child of degree 3. By Proposition 1, y has B 2 -branches as its children, which is a contradiction to the hypothesis that u is the unique vertex in T having B 2 -branches as its children. Now suppose that u is not the root vertex of T . Denote by v the parent vertex of u in T . Suppose that every branch attached to u is a B k -branch. From Theorem 2 and Lemma 3, k = 2 or 3, i.e., T is of the form as depicted in Fig. 2. Now by Lemmas 9 and 13, we get that u has at most two B 2 -branches as its children, i.e., T contains at most two B 2 -branches. Otherwise, there is a child of u, say z, which has a child of degree more than 2, then by Proposition 1, z has B 2 -branches as its children, which is a contradiction to the hypothesis that u is the unique vertex in T having B 2 -branches as its children.
Now the result follows.
In particular, if a minimal-ABC tree contains exactly two B 2 -branches, then the two B 2branches are attached to the same parent, and their siblings can only be B 3 -branches.

Conclusion
With the results presented in this work, we made an additional step towards the complete characterization of minimal-ABC trees, one of the most known and most difficult open problems in mathematical chemistry. Namely, it is known that a minimal-ABC tree cannot contain more than eleven B 2 -branches. Here we show that if a minimal-ABC tree of order larger than 39 contains B 1 -branches, then it contains exactly one B 2 -branch, and if a minimal-ABC tree of order larger than 163 contains no B 1 -branch, then it contains at most two B 2 -branches. We also strongly believe that after some not very large size, there are also no B 2 -branches in a minimal-ABC tree. To determine when it happens remains as an open problem for a future research.