The minimal-ABC trees with B1-branches

Darko Dimitrov; Zhibin Du; Carlos M. da Fonseca

doi:10.1371/journal.pone.0195153

Abstract

The atom-bond connectivity index (or, for short, ABC index) is a molecular structure descriptor bridging chemistry to graph theory. It is probably the most studied topological index among all numerical parameters of a graph that characterize its topology. For a given graph G = (V, E), the ABC index of G is defined as , where d_i denotes the degree of the vertex i, and ij is the edge incident to the vertices i and j. A combination of physicochemical and the ABC index properties are commonly used to foresee the bioactivity of different chemical composites. Additionally, the applicability of the ABC index in chemical thermodynamics and other areas of chemistry, such as in dendrimer nanostars, benzenoid systems, fluoranthene congeners, and phenylenes is well studied in the literature. While finding of the graphs with the greatest ABC-value is a straightforward assignment, the characterization of the tree(s) with minimal ABC index is a problem largely open and has recently given rise to numerous studies and conjectures. A B₁-branch of a graph is a pendent path of order 2. In this paper, we provide an important step forward to the full characterization of these minimal trees. Namely, we show that a minimal-ABC tree contains neither 4 nor 3 B₁-branches. The case when the number of B₁-branches is 2 is also considered.

Citation: Dimitrov D, Du Z, Fonseca CMd (2018) The minimal-ABC trees with B₁-branches. PLoS ONE 13(4): e0195153. https://doi.org/10.1371/journal.pone.0195153

Editor: L. Michel Espinoza-Fonseca, University of Michigan, UNITED STATES

Received: August 23, 2017; Accepted: February 14, 2018; Published: April 18, 2018

Copyright: © 2018 Dimitrov et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All relevant data are within the paper.

Funding: This work was supported by the National Natural Science Foundation of China (grant number 11701505 to Dr. Zhibin Du) and the Guangdong Provincial Natural Science Foundation of China (grant number 2014A030310277 to Dr. Zhibin Du). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Competing interests: The authors have declared that no competing interests exist.

Introduction

The atom-bond connectivity index, widely known as ABC index, of a graph is a thoroughly studied vertex-degree-based graph invariant both in chemistry and mathematical communities. 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. The atom-bond connectivity index (or, simply, ABC index) is a vertex-degree-based graph topological index, which is a variation of the Randić graph-theoretic invariant [1], and is defined as where

The relevance of the ABC index, in what we call today chemical graph theory, was first revealed two decades ago by Estrada, Torres, Rodríguez, and Gutman in [2]. They disclosed the importance of the ABC index as an analytical instrument for modeling thermodynamic properties of organic chemical compounds. Ten years later, Estrada [3] uncovered the significance of ABC index on the stability of branched alkanes, based on at that time a novel quantum-theory-like exposition. These studies were the trigger point for an uncountable number of papers on a new found area: chemical graph theory. Just to give two examples, in [4] it is proved that the ABC index of both benzenoid systems and fluoranthene congeners, consisting of two benzenoid fragments, depend exclusively on the number of vertices, hexagons and inlets. The author also characterized the extremal catacondensed benzenoid systems with the maximal and minimal ABC indices. The case of the phenylenes was considered by [5]. Another example of the importance of this topological descriptor can be seen on the calculation of the ABC index of an infinite class of nanostar dendrimers, artificially manufactured or synthesized molecule built up from branched units called monomers [6].

Many problems persist open, though. For example, it is known that the star of a given order has the maximal ABC index [7]. However for the trees with minimal ABC index, we are still far from a full characterization. For some further conjectures and partial results the reader is referred to [8–12]. More progress about minimal ABC trees can be found in [13–18].

A path v₀v₁⋯v_r in a graph G is said to be a pendent path of length r, where d_v₀ ≥ 3, d_v₁ = ⋯ = d_{v_r−1} = 2, and d_{v_r} = 1.

For the tree(s) with minimal ABC index, the length of its pendent paths is of crucial importance. In particular, the next lemma has become a key result in this area:

Lemma 1 [11, 19] If T is a tree with minimal ABC index, then every pendent path in T is of length 2 or 3, and there is at most one pendent path of length 3 in T.

In [20], Wang defined the greedy trees, for a given degree sequence, as follows:

Definition 1. Suppose that the degrees of the non-leaf vertices are given, the greedy tree is achieved by the following ‘greedy algorithm’:

Label the vertex with the largest degree as v (the root);
Label the neighbors of v as v₁, v₂, …, assign the largest degree available to them such that d(v₁) ≥ d(v₂) ≥ ⋯;
Label the neighbors of v₁ (except v) as v₁₁, v₁₂, …, such that they take all the largest degrees available and that d(v₁₁) ≥ d(v₁₂) ≥ ⋯, then do the same for v₂, v₃, …;
Repeat (3) for all newly labeled vertices, always starting with the neighbors of the labeled vertex with largest degree whose neighbors are not labeled yet.

In particular, the vertex i is said to be the root of T, which is also the vertex lying on the first layer of T; the vertices i₁, i₂, … are said to be the vertices lying on the second layer of T; the vertices i₁₁, i₁₂, … are said to be the vertices lying on the third layer of T, and so on.

A major result attesting the importance of the greedy trees is the next proposition.

Proposition 2 ([21, 22]). Given the degree sequence, the greedy tree minimizes the ABC index.

From the previous considerations, different types of branches will play a crucial role in our quest. Namely, the B_k-branches, with k ≥ 1, and the -branches, with k ≥ 1, are illustrated in Fig 1.

Download:

Fig 1. The B_k- and

-branches for k ≥ 1.

https://doi.org/10.1371/journal.pone.0195153.g001

In this regard, the most relevant results on minimal-ABC trees are listed next.

Proposition 3 ([23, Theorem 3.2]). A minimal-ABC tree does not contain B_k-branch, with k > 4.

Proposition 4 ([24, Proposition 3.4]). A minimal-ABC tree does not contain a B₃-branch and a -branch sharing a common parent vertex.

Proposition 5 ([23, Lemma 3.3(a)]). A minimal-ABC tree does not contain a B₄-branch and a B₁-branch sharing a common parent vertex.

Proposition 6 ([25, Theorem 3.4]). A minimal-ABC tree of order n > 18 with a pendent path of length 3 may contain a B₂-branch if and only if it is of order 161 or 168. Moreover, in this case, a minimal-ABC tree is comprised of a single central vertex, B₃-branches and one B₂, including a pendent path of length 3 that may belong to a -branch or -branch.

As a consequence of Proposition 6, we get the following proposition immediately.

Proposition 7. A minimal-ABC tree cannot contain a B₂-branch and a -branch simultaneously.

Recently, the authors were able to show in [26] that a minimal-ABC tree cannot contain simultaneously a B₄-branch and B₁- or B₂-branches.

Recall that a k-terminal vertex of a rooted tree is a vertex of degree k + 1 ≥ 3, which is a parent of only B_≥1-branches, such that at least one branch among them is a B₁-branch (or -branch). The (sub)tree, induced by a k-terminal vertex and all its (direct and indirect) children (descendant) vertices, is called a k-terminal branch or T_k-branch.

Proposition 8 ([27, Proposition 2.13]). A minimal-ABC tree contains at most one T_k-branch, with k ≥ 2.

Proposition 9 ([27, Theorem 3.5]). A minimal-ABC tree contains at most four B₁-branches.

Although all the progress that has been lately made, the minimal-ABC trees seem still far from a full characterization. This paper contributes for this task. Specifically, we show that such trees contain neither 4 nor 3 B₁-branches. The case when we have 2 B₁-branches is also considered in the last section.

Preliminaries and methods

Lemmas

First we recall some technical lemmas.

Lemma 10 ([23, Proposition A.3]). Let with real numbers x, y ≥ 2, Δx ≥ 0, 0 ≤ Δy < 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 10.

Lemma 11. Let with real numbers x, y ≥ 2, 0 ≤ Δx < x, Δy ≥ 0. Then g(x, y) decreases in x and increases in y.

In a similar fashion we have:

Lemma 12. Let h(x, y) = (y − 4)f(x + y − 5, 4) − f(x, y), where x ≥ y and y = 6, 7, 8, 9, 10, 11. Then for every fixed y, the function h(x, y) decreases in x ≥ y.

proof. We only prove the case when y = 6. The other cases are similar.

Suppose that y = 6. Then h(x, 6) = 2f(x + 1, 4) − f(x, 6).

First we have Next, it is readily verified that for x ≥ 6.

Now it follows that h′(x, 6) < 0, i.e., h(x, 6) decreases in x ≥ 6.

Similar to the proof of Lemma 12, we can also get the following lemma.

Lemma 13. Let ℓ(x, y) = (y − 3)f(x + y − 4, 3) − f(x, y), where y = 5, 7, 8, 9.

When y = 5, the function ℓ(x, 5) increases in x > 0.
When y = 7, the function ℓ(x, 7) decreases in x ≥ 19.
When y = 8, the function ℓ(x, 8) decreases in x ≥ 17.
When y = 9, the function ℓ(x,9) decreases in x ≥ 16.

The root of B₁-branches

A Kragujevac tree is a tree comprising of a single central vertex, B_k-branches, with k ≥ 1, and at most one -branch.

Lemma 14 ([28, Theorem 11]). If T is a Kragujevac tree with minimal ABC index, and the degree of the central vertex of T is at least 19, then T contains no B₁-branch.

Taking into account Lemma 14, we can establish the main result in this section.

Proposition 15. If T is a minimal-ABC tree on more than 122 vertices containing B₁-branches, then the B₁-branches cannot be attached to the root vertex of T.

proof. Observe that the B₁-branches of T are attached to the same vertex, say u, otherwise, there are at least two T_k-branches, which is a contradiction to Proposition 8. Suppose to the contrary that u is the root vertex of T.

First, by Proposition 3, u contains no B_k-branch with k > 4. Next by Proposition 5, u contains no B₄-branch, and by Propositions 4 and 7, u contains no -branch, no matter u has B₃-branches or B₂-branches. Now we may deduce that the branches attached to u must be B₃-, B₂- or B₁-branches, i.e., T is of the structure as depicted in Fig 2.

Download:

Fig 2. The tree T in the proof of Proposition 15.

https://doi.org/10.1371/journal.pone.0195153.g002

Notice that T is actually a Kragujevac tree. Denote by d_u the degree of u in T.

If d_u ≥ 19, then from Lemma 14, T contains no B₁-branch, which is a contradiction to the assumption for the existence of B₁-branches in T.

If d_u ≤ 18, then recall that every branch attached to u in T is a B_k-branch with k = 1, 2, 3, and thus the order of T is at most which is a contradiction to the assumption for the order of T.

Now the result follows.

Since all the minimal-ABC trees of order up to 300 are completely determined in [29], we may assume that the trees considered in our main results have more than 300 vertices.

Switching transformation

Before we proceed with the main results of this paper, we present the so-called switching transformation explicitly stated by Lin, Gao, Chen, and Lin [30].

Lemma 16 (Switching transformation). Let G = (V, E) be a connected graph with uv, xy ∈ E(G) and uy, xv ∉ E(G). Let G₁ = G − uv − xy + uy + xv. If d(u) ≥ d(x) and d(v) ≤ d(y), then ABC(G₁) ≤ ABC(G), with the equality if and only if d(u) = d(x) or d(v) = d(y).

The switching transformation was used in the proofs of some characterizations of the minimal-ABC trees, and the following observation that will be applied in the further analysis.

Observation 1. Let G be a minimal-ABC tree with the root vertex v₀ and let v₀, v₁, …, v_n be the sequence of vertices obtain by the breadth-first search of G. If d(v_i), d(v_j) ≥ 3 and i < j, then by Lemma 16, we may assume that d(v_i) ≥ d(v_j).

From Observation 1, we may assume that the trees considered are all greedy trees.

Results

The existence of four B₁-branches

In this section we will prove our first main result: Any minimal-ABC tree cannot contain four B₁-branches.

The following result is recent and establishes a forbidden configuration for minimal-ABC trees.

Proposition 17 ([27, Proposition 3.2]). When s + t > 6, the configuration T depicted in Fig 3 cannot occur in a minimal-ABC tree.

Download:

Fig 3. The tree T in Proposition 17 and Theorem 18.

https://doi.org/10.1371/journal.pone.0195153.g003

We are ready now to state the main result of this section.

Theorem 18. A minimal-ABC tree cannot contain four B₁-branches.

proof. Suppose to the contrary that T is a minimal-ABC tree containing exactly four B₁-branches. Observe that the four B₁-branches are attached to the same vertex, say u, otherwise, there are at least two T_k-branches, which is a contradiction to Proposition 8. Moreover, by Proposition 15, u is not the root vertex of T. Let us denote by v the parent of u.

First, by Proposition 3, u contains no B_k-branch with k > 4. Next by Proposition 5, u contains no B₄-branch, and by Propositions 4 and 7, u contains no -branch, no matter u has B₃-branches or B₂-branches. Now we may deduce that the branches attached to u must be B₃-, B₂- or B₁-branches, i.e., T is of the structure depicted in Fig 3.

Denote by s the number of B₃-branches attached to u, and t the number of B₂-branches attached to u. Clearly, s + t ≥ 1, and s + t ≤ 6 from Proposition 17.

Let d_x be the degree of vertex x in T.

Observe that d_v ≥ d_u = s + t + 5 from Proposition 2.

Case 1. t = 0.

In this case, we apply the transformation depicted in Fig 4.

Download:

Fig 4. The transformation

in Case 1 of Theorem 18.

https://doi.org/10.1371/journal.pone.0195153.g004

After applying , the degree of vertex v increases by s, while the degree of vertex u decreases by s. The rest of the vertices do not change their degrees. The change of the ABC index after applying is

Clearly, f(d_v + s, d_x) − f(d_v, d_x) < 0 for , and thus

Recall that d_v ≥ s + 5 from Proposition 2. On one hand, by Lemma 12, (s + 1)f(d_v + s, 4) − f(d_v, s + 5) decreases in d_v ≥ s + 5. On the other hand, by Lemma 11, f(d_v + s, 5) − f(d_v + s, 4) also decreases in d_v ≥ s + 5. So we get that (1) By virtue of Mathematica, the right-hand side of (1) is negative, equivalently ABC(T₁) < ABC(T), follows from direct calculation, for 1 ≤ s ≤ 6.

Case 2. t ≥ 1.

In this case, we apply the transformation depicted in Fig 5.

Download:

Fig 5. The transformation

in Case 2 of Theorem 18.

https://doi.org/10.1371/journal.pone.0195153.g005

After applying , the degree of vertex v increases by s + t, while the degree of vertex u decreases to 4, and a child of u in T belonging to a B₂-branch increases its degree from 3 to 4. The rest of the vertices do not change their degrees. The change of the ABC index after applying is (2)

Clearly, f(d_v + s + t, d_x) − f(d_v, d_x) < 0 for , and thus

Let r = s + t be a fixed number. Recall that 1 ≤ r ≤ 6.

Now we have (3) For the right-hand side of (3), notice that the coefficient of t is Since d_v > 5, from Lemma 10, f(d_v + r, y) − f(r + 5, y) decreases in y ≥ 2, thus we may deduce that Together with t ≤ r, we have (4)

Recall that d_v ≥ r + 5 from Proposition 2.

Subcase 2.1. r = 1.

If r = 1, then by (4), we have Moreover, by Lemma 12, we know that 2f(d_v + 1, 4) − f(d_v, 6) decreases in d_v ≥ 6, thus i.e., ABC(T₁) < ABC(T).

Subcase 2.2. r = 2.

If r = 2, then by (4), we have Moreover, by Lemma 12, we know that 3f(d_v + 2, 4) − f(d_v, 7) decreases in d_v ≥ 7, and by Lemma 10, f(d_v + 2, 3) − f(d_v + 2, 4) increases in d_v, thus

So for d_v ≥ 11, we get that i.e., ABC(T₁) < ABC(T). For the remaining cases that 7 ≤ d_v ≤ 10, by virtue of Mathematica, the right-hand side of (4) is negative, equivalently ABC(T₁) < ABC(T), follows from direct calculation easily.

Subcase 2.3. r = 3.

If r = 3, then by (4), we have Moreover, by Lemma 12, we know that 4f(d_v + 3, 4) − f(d_v, 8) decreases in d_v ≥ 8, and by Lemma 10, f(d_v + 3, 3) − f(d_v + 3, 4) increases in d_v, thus

So for d_v ≥ 20, we get that i.e., ABC(T₁) < ABC(T). For the remaining cases that 8 ≤ d_v ≤ 19, by virtue of Mathematica, the right-hand side of (4) is negative, equivalently ABC(T₁) < ABC(T), follows from direct calculation easily.

Subcase 2.4. r = 4.

If r = 4, then by (4), we have Moreover, by Lemma 12, we know that 5f(d_v + 4, 4) − f(d_v, 9) decreases in d_v ≥ 9, and by Lemma 10, f(d_v + 4, 3) − f(d_v + 4, 4) increases in d_v, thus

So for d_v ≥ 31, we get that i.e., ABC(T₁) < ABC(T). For the remaining cases that 9 ≤ d_v ≤ 30, by virtue of Mathematica, the right-hand side of (4) is negative, equivalently ABC(T₁) < ABC(T), follows from direct calculation easily.

Subcase 2.5. r = 5.

If r = 5, then by (4), we have Moreover, by Lemma 12, we know that 6f(d_v + 5, 4) − f(d_v, 10) decreases in d_v ≥ 10, and by Lemma 10, f(d_v + 5, 3) − f(d_v + 5, 4) increases in d_v, thus

So for d_v ≥ 42, we get that i.e., ABC(T₁) < ABC(T). For the remaining cases that 10 ≤ d_v ≤ 41, by virtue of Mathematica, the right-hand side of (4) is negative, equivalently ABC(T₁) < ABC(T), follows from direct calculation easily.

Subcase 2.6. r = 6.

If r = 6, then by (4), we have Moreover, by Lemma 12, we know that 7f(d_v + 6, 4) − f(d_v, 11) decreases in d_v ≥ 11, and by Lemma 10, f(d_v + 6, 3) − f(d_v + 6, 4) increases in d_v, thus

So for d_v ≥ 56, we get that i.e., ABC(T₁) < ABC(T). For the cases that 18 ≤ d_v ≤ 55, by virtue of Mathematica, the right-hand side of (4) is negative, equivalently ABC(T₁) < ABC(T), follows from direct calculation easily.

As to the remaining cases that 11 ≤ d_v ≤ 17, let us be a bit more precisely in (2) about the term Notice that the degree of every neighbor of v in is at least 3 from Proposition 2. Furthermore, by Lemma 10, f(d_v + 6, d_x) − f(d_v, d_x) decreases in d_x ≥ 3, we may deduce that Now together with (4), it follows that (5) By virtue of Mathematica, the right-hand side of (5) is negative, equivalently ABC(T₁) < ABC(T), for 11 ≤ d_v ≤ 17, follows from direct calculation easily.

Combining the above cases, the result follows easily.

The existence of three B₁-branches

We proceed proving in this section that a minimal-ABC tree does not contain three B₁-branches. Before that, we consider some preliminary results.

Proposition 19 ([27, Proposition 3.2]). When s + t > 8, the configuration T depicted in Fig 6 cannot occur in a minimal-ABC tree.

Download:

Fig 6. The tree T in Propositions 19, 20 and 21, and Theorem 22.

https://doi.org/10.1371/journal.pone.0195153.g006

Proposition 20 ([27, Proposition 3.4]). When s = 0 and t > 3, the configuration T depicted in Fig 6 cannot occur in a minimal-ABC tree.

Proposition 21. The configuration T depicted in Fig 6 cannot occur in a minimal-ABC tree, for the following cases:

t = 3 and s = 0, 4, 5;
t = 4 and s = 2, 3, 4;
t = 5 and s = 1, 2, 3;
t = 6 and s = 1, 2;
t = 7 and s = 1.

proof. Let d_x be the degree of vertex x in T.

First we apply the transformation illustrated in Fig 7.

Download:

Fig 7. The transformation

in the proof of Proposition 21.

https://doi.org/10.1371/journal.pone.0195153.g007

After applying , the degree of vertex u decreases by 3, while the degrees of three children of u in T belonging to a B₂-branch increase from 3 to 4, and the rest of the vertices do not change their degrees. The change of the ABC index after applying is

From Lemma 11, f(s + t + 1, d_v) − f(s + t + 4, d_v) increases in d_v, and thus Now it follows that (6) By virtue of Mathematica, the right-hand side of (6) is negative, equivalently ABC(T₁) < ABC(T), follows from direct calculation easily, except the case t = 3 and s = 0. In such case, we apply the transformation illustrated in Fig 8.

Download:

Fig 8. The transformation

in the proof of Proposition 21.

https://doi.org/10.1371/journal.pone.0195153.g008

After applying , the degree of vertex v increases by 2, the degrees of three children of u in T belonging to a B₂-branch increase from 3 to 4, a pendent vertex in T belonging to a B₂-branch increases its degree from 1 to 2, the degree of u decreases from 7 to 1, and the rest of the vertices do not change their degrees. The change of the ABC index after applying is (7)

Clearly, f(d_v + 2, d_x) − f(d_v, d_x) < 0 for . So

Note that d_v ≥ d_u = 7 from Proposition 2, and from Lemma 12, we know that 3f(d_v + 2, 4) − f(d_v, 7) decreases in d_v ≥ 7. Therefore, for d_v ≥ 20, we get that

For the remaining cases that 7 ≤ d_v ≤ 19, let us be a bit more precisely in (7) for the term Note that every neighbor of v in has degree at least three from Proposition 2. By Lemma 10, f(d_v + 2, d_x) − f(d_v, d_x) decreases in d_x ≥ 3, and thus Now together with (7), it follows that (8) By virtue of Mathematica, the right-hand side of (8) is negative, equivalently ABC(T₁) < ABC(T), for 7 ≤ d_v ≤ 19, follows from direct calculation easily.

Then the result follows.

We are now prepared to establish the main result of this section.

Theorem 22. A minimal-ABC tree cannot contain three B₁-branches.

proof. Similarly to Theorem 18, let us suppose to the contrary that T is a minimal-ABC tree containing exactly three B₁-branches. Observe that the three B₁-branches are attached to the same vertex, say u, otherwise, there are at least two T_k-branches, which is a contradiction to Proposition 8. Moreover, by Proposition 15, u is not the root vertex of T. Denote by v the parent of u.

First, by Proposition 3, u contains no B_k-branch with k > 4. Next by Proposition 5, u contains no B₄-branch, and by Propositions 4 and 7, u contains no -branch, no matter u has B₃-branches or B₂-branches. Now we may deduce that the branches attached to u must be B₃-, B₂- or B₁-branches, i.e., T is of the structure depicted in Fig 6.

Let us denote by s the number of B₃-branches attached to u, and by t the number of B₂-branches attached to u. Clearly, s + t ≥ 1, and s + t ≤ 8, from Proposition 19.

We apply the transformation depicted in Fig 9. And let d_x be the degree of vertex x in T.

Download:

Fig 9. The transformation

in the proof of Theorem 22.

https://doi.org/10.1371/journal.pone.0195153.g009

After applying , the degree of vertex v increases by s + t, while the degree of vertex u decreases by s + t, and the rest of the vertices do not change their degrees. The change of the ABC index after applying is (9)

Clearly, f(d_v + s + t, d_x) − f(d_v, d_x) < 0 for , and thus

On one hand, from Lemma 10, f(d_v + s + t, 4) − f(d_v, s + t + 4) increases in d_v, thus So it follows that (10)

On the other hand, note that d_v ≥ d_u = s + t + 4 from Proposition 2, and both f(d_v + s + t, 4) and f(d_v + s + t, 3) decrease in d_v ≥ s + t + 4, i.e., the right-hand side of (10) also decreases in d_v ≥ s + t + 4.

Besides the upper bound about ABC(T₁) − ABC(T) as (10), by considering a bit precisely in (9) for the term we may get a somewhat stricter upper bound about ABC(T₁) − ABC(T). Note that, from Lemma 10, f(d_v + s + t, d_x) − f(d_v, d_x) decreases in d_x, and from Proposition 2, every neighbor of v in has degree at least three, thus Now together with (9), it follows that (11)

Case 1. t = 0.

In this case, note that 1 ≤ s ≤ 8, and d_v ≥ s + 4.

By direct calculation, we may deduce that the right-hand side of (10) is negative, equivalently ABC(T₁) < ABC(T), holds for the following cases:

s = 1 and d_v ≥ 12;
s = 2 and d_v ≥ 14;
s = 3 and d_v ≥ 16;
s = 4 and d_v ≥ 18;
s = 5 and d_v ≥ 21;
s = 6 and d_v ≥ 23;
s = 7 and d_v ≥ 26;
s = 8 and d_v ≥ 26.

For the remaining cases as follows:

s = 1 and 5 ≤ d_v ≤ 11;
s = 2 and 6 ≤ d_v ≤ 13;
s = 3 and 7 ≤ d_v ≤ 15;
s = 4 and 8 ≤ d_v ≤ 17;
s = 5 and 9 ≤ d_v ≤ 20;
s = 6 and 10 ≤ d_v ≤ 22;
s = 7 and 11 ≤ d_v ≤ 25;
s = 8 and 12 ≤ d_v ≤ 25,

we would turn to use (11), and negative upper bounds, equivalently ABC(T₁) < ABC(T), follow from direct calculation.

Case 2. t = 1.

In this case, note that 0 ≤ s ≤ 7, and d_v ≥ s + 5.

By direct calculation, we may deduce that the right-hand side of (10) is negative, equivalently ABC(T₁) < ABC(T), holds for the following cases:

s = 0 and d_v ≥ 124;
s = 1 and d_v ≥ 23;
s = 2 and d_v ≥ 22;
s = 3 and d_v ≥ 22;
s = 4 and d_v ≥ 25;
s = 5 and d_v ≥ 28;
s = 6 and d_v ≥ 30;
s = 7 and d_v ≥ 33.

For the remaining cases as follows:

s = 0 and 5 ≤ d_v ≤ 123;
s = 1 and 6 ≤ d_v ≤ 22;
s = 2 and 7 ≤ d_v ≤ 21;
s = 3 and 8 ≤ d_v ≤ 21;
s = 4 and 9 ≤ d_v ≤ 24;
s = 5 and 10 ≤ d_v ≤ 27;
s = 6 and 11 ≤ d_v ≤ 29;
s = 7 and 12 ≤ d_v ≤ 32,

we would turn to use (11), and negative upper bounds, equivalently ABC(T₁) < ABC(T), follow from direct calculation easily.

Case 3. t = 2.

In this case, note that 0 ≤ s ≤ 6, and d_v ≥ s + 6.

By direct calculation, we may deduce that the right-hand side of (10) is negative, equivalently ABC(T₁) < ABC(T), holds for the following cases:

s = 0 and d_v ≥ 751;
s = 1 and d_v ≥ 41;
s = 2 and d_v ≥ 34;
s = 3 and d_v ≥ 33;
s = 4 and d_v ≥ 35;
s = 5 and d_v ≥ 37;
s = 6 and d_v ≥ 39.

For the remaining cases as follows:

s = 0 and 6 ≤ d_v ≤ 750;
s = 1 and 7 ≤ d_v ≤ 40;
s = 2 and 8 ≤ d_v ≤ 33;
s = 3 and 9 ≤ d_v ≤ 32;
s = 4 and 10 ≤ d_v ≤ 34;
s = 5 and 11 ≤ d_v ≤ 36;
s = 6 and 12 ≤ d_v ≤ 38,

we would turn to use (11), and negative upper bounds, equivalently ABC(T₁) < ABC(T), follow from direct calculation easily.

Case 4. t = 3.

In this case, note that 0 ≤ s ≤ 5, and d_v ≥ s + 7.

On one hand, the contradiction for the cases s = 0, 4, 5 may be deduced from Proposition 21.

On the other hand, by direct calculation, we may deduce that the right-hand side of (10) is negative, equivalently ABC(T₁) < ABC(T), holds for the following cases:

s = 1 and d_v ≥ 74;
s = 2 and d_v ≥ 52;
s = 3 and d_v ≥ 48.

For the remaining cases as follows:

s = 1 and 8 ≤ d_v ≤ 73;
s = 2 and 9 ≤ d_v ≤ 51;
s = 3 and 10 ≤ d_v ≤ 47,

we would turn to use (11), and negative upper bounds, equivalently ABC(T₁) < ABC(T), follow from direct calculation easily.

Case 5. t = 4.

In this case, note that 0 ≤ s ≤ 4, and d_v ≥ s + 8.

The contradiction for the cases s = 0 and s = 2, 3, 4 may be, respectively, deduced from Propositions 20 and 21.

Besides that, by direct calculation, we may deduce that the right-hand side of (10) is negative, equivalently ABC(T₁) < ABC(T), for s = 1 and d_v ≥ 145. For the remaining cases s = 1 and 9 ≤ d_v ≤ 144, we would turn to use (11), and a negative upper bound, equivalently ABC(T₁) < ABC(T), follows from direct calculation easily.

Case 6. t = 5.

In this case, note that s = 0, 1, 2, 3. The contradiction for the cases s = 0 and s = 1, 2, 3 may be, respectively, deduced from Propositions 20 and 21.

Case 7. t = 6.

In this case, note that s = 0, 1, 2. The contradiction for the cases s = 0 and s = 1, 2 may be, respectively, deduced from Propositions 20 and 21.

Case 8. t = 7.

In this case, note that s = 0, 1. The contradiction for the cases s = 0 and s = 1 may be, respectively, deduced from Propositions 20 and 21.

Case 9. t = 8.

In this case, note that s = 0. The contradiction may be deduced from Proposition 20 directly.

Combining the above cases, the result follows.

The existence of two B₁-branches

This last section is devoted to the analysis of the existence of two B₁-branches in a minimal-ABC tree. The first two propositions are known results establishing forbidden configurations in such cases.

Proposition 23 ([27, Proposition 3.2]). When s + t > 10, the configuration T depicted in Fig 10 cannot occur in a minimal-ABC tree.

Download:

Fig 10. The tree T in Propositions 23, 24 and 25, and Theorem 26.

https://doi.org/10.1371/journal.pone.0195153.g010

Proposition 24 ([27, Proposition 3.4]). When s = 0 and t > 4, the configuration T depicted in Fig 10 cannot occur in a minimal-ABC tree.

We next list several cases more where the configuration depicted in Fig 10 is not possible in a minimal-ABC tree.

Proposition 25. The configuration T depicted in Fig 10 cannot occur in a minimal-ABC tree, for the following cases:

t = 2 and s = 0;
t = 3 and s = 1, 2;
t = 4 and s = 0, 1, 2, 3, 4, 5, 6;
t = 5 and s = 1, 2, 3, 4, 5;
t = 6 and s = 1, 2, 3, 4;
t = 7 and s = 1, 2, 3;
t = 8 and s = 1, 2;
t = 9 and s = 1.

proof. First we apply the transformation illustrated in Fig 11. Let d_x be the degree of vertex x in T.

Download:

Fig 11. The transformation

in the proof of Proposition 25.

https://doi.org/10.1371/journal.pone.0195153.g011

After applying , the degree of vertex u decreases by 2, while the degrees of two children of u in T belonging to a B₂-branch increase from 3 to 4. The rest of the vertices do not change their degrees. The change of the ABC index after applying is

From Lemma 11, f(s + t + 1, d_v) − f(s + t + 3, d_v) increases in d_v, and thus

Now it follows that (12) The right-hand side of (12) is negative, equivalently ABC(T₁) < ABC(T), holds for the following cases:

t = 4 and s = 3, 4, 5, 6;
t = 5 and s = 2, 3, 4, 5;
t = 6 and s = 1, 2, 3, 4;
t = 7 and s = 1, 2, 3;
t = 8 and s = 1, 2;
t = 9 and s = 1.

Next for the following cases:

t = 2 and s = 0;
t = 3 and s = 1, 2;
t = 4 and s = 1, 2,

we apply the transformation illustrated in Fig 12.

Download:

Fig 12. The transformation

in the proof of Proposition 25.

https://doi.org/10.1371/journal.pone.0195153.g012

After applying , the degree of vertex v increases by s + t − 1, the degrees of two children of u in T belonging to a B₂-branch increase from 3 to 4, a pendent vertex in T belonging to a B₃-branch increases its degree from 1 to 2, the degree of u decreases from s + t + 3 to 1, and the rest of the vertices do not change their degrees. The change of the ABC index after applying is (13)

Clearly, f(d_v + s + t − 1, d_x) − f(d_v, d_x)≤0, for . So (14)

Note that d_v ≥ d_u = s + t + 3 from Proposition 2, and from Lemma 13, we know that (s + t)f(d_v + s + t − 1, 3) − f(d_v, s + t + 3) increases in d_v ≥ 5 when t = 2 and s = 0, and decreases in

d_v ≥ 19 when t = 3 and s = 1;
d_v ≥ 17 when t = 3 and s = 2, or t = 4 and s = 1;
d_v ≥ 16 when t = 4 and s = 2.

On the other hand, from Lemma 11, f(d_v + s + t − 1, 4) − f(d_v + s + t − 1, 3) also decreases in d_v ≥ s + t + 3.

So if t = 2 and s = 0, and d_v ≥ 83, then by (14), Otherwise, the right-hand of (14) decreases in the following cases:

d_v ≥ 19 when t = 3 and s = 1;
d_v ≥ 17 when t = 3 and s = 2 or t = 4 and s = 1;
d_v ≥ 16 when t = 4 and s = 2.

Besides the upper bound about ABC(T₁) − ABC(T) as (14), by considering in particular in (13) the term we may get a somewhat stricter upper bound about ABC(T₁) − ABC(T). Note that, from Lemma 10, f(d_v + s + t − 1, d_x) − f(d_v, d_x) decreases in d_x, and from Proposition 2, every neighbor of v in has degree at least three, thus Now together with (13), it follows that (15)

By direct calculation, we may deduce that the right-hand side of (14) is negative, equivalently ABC(T₁) < ABC(T), holds for the following cases:

t = 3, s = 1, and d_v ≥ 64;
t = 3, s = 2, and d_v ≥ 44;
t = 4, s = 1, and d_v ≥ 4015;
t = 4, s = 2, and d_v ≥ 116.

For the remaining cases as follows:

t = 2, s = 0, and 5 ≤ d_v ≤ 82;
t = 3, s = 1, and 7 ≤ d_v ≤ 63;
t = 3, s = 2, and 8 ≤ d_v ≤ 43;
t = 4, s = 1, and 8 ≤ d_v ≤ 4014;
t = 4, s = 2, and 9 ≤ d_v ≤ 115,

we would turn to use (15), and negative upper bounds, equivalently ABC(T₁) < ABC(T), follow from direct calculation easily.

At this point, there are still two remaining cases: t = 4, s = 0, and t = 5, s = 1.

For the case t = 4 and s = 0, we apply the transformation illustrated in Fig 13.

Download:

Fig 13. The transformation

in the proof of Proposition 25.

https://doi.org/10.1371/journal.pone.0195153.g013

After applying , the degree of vertex v increases by 2, the degrees of two children of u in T belonging to a B₂-branch increase from 3 to 5, one child of u in T belonging to another B₂-branch increases its degree from 3 to 4, the remaining child of u in T belonging to a B₂-branch decreases its degree from 3 to 2, the degree of u decreases from 7 to 1, and the rest of the vertices do not change their degrees. The change of the ABC index after applying is (16)

Clearly, for . So (17)

Note that d_v ≥ d_u = 7 from Proposition 2, and by Lemma 11, f(d_v + 2, 5) − f(d_v + 2, 4) decreases in d_v ≥ 7. On the other hand, by Lemma 12, 3f(d_v + 2, 4) − f(d_v, 7) decreases in d_v ≥ 7. So the right-hand side of (17) also decreases in d_v ≥ 7.

Besides the upper bound about ABC(T₁) − ABC(T) as (17), by considering in (16) the term we may get a somewhat stricter upper bound about ABC(T₁) − ABC(T). Note that, from Lemma 10, f(d_v + 2, d_x) − f(d_v, d_x) decreases in d_x, and from Proposition 2, every neighbor of v in has degree at least three, thus Now together with (16), it follows that (18)

For d_v ≥ 20, by (17), we have i.e., ABC(T₁) < ABC(T). For the remaining cases 7 ≤ d_v ≤ 19, we would turn to use (18), and a negative upper bound, equivalently ABC(T₁) < ABC(T), follows from direct calculation straightforwardly.

As to the last case t = 5 and s = 1, we apply the transformation illustrated in Fig 14.

Download:

Fig 14. The transformation

in the proof of Proposition 25.

https://doi.org/10.1371/journal.pone.0195153.g014

After applying , the degree of vertex v increases by 4, three children of u in T belonging to a B₂-branch increase its degrees from 3 to 4, the degree of one child of u in T belonging to another B₂-branch increases from 3 to 5, the remaining child of u in T belonging to a B₂-branch decreases its degree from 3 to 2, the degree of u decreases from 9 to 1, and the rest of the vertices do not change their degrees. The change of the ABC index after applying is (19)

Clearly, f(d_v + 4, d_x) − f(d_v, d_x) < 0 for . So (20)

Note that d_v ≥ d_u = 9 from Proposition 2, and by Lemma 11, f(d_v + 4, 5) − f(d_v + 4, 4) decreases in d_v ≥ 9. On the other hand, by Lemma 12, 5f(d_v + 4, 4) − f(d_v, 9) decreases in d_v ≥ 9. So the right-hand side of (20) also decreases in d_v ≥ 9.

Besides the upper bound about ABC(T₁) − ABC(T) as (20), by considering in (19) the term we may get a somewhat stricter upper bound about ABC(T₁) − ABC(T). Note that, from Lemma 10, f(d_v + 4, d_x) − f(d_v, d_x) decreases in d_x, and from Proposition 2, every neighbor of v in has degree at least three, thus Now together with (19), it follows that (21)

For d_v ≥ 15, by (20), we have i.e., ABC(T₁) < ABC(T). For the remaining cases 9 ≤ d_v ≤ 14, we would turn to use (21), and a negative upper bound, equivalently ABC(T₁) < ABC(T), follows from direct calculation easily.

Combining the above arguments, the result follows.

Our main result is stated next. As we will see, the configuration depicted in Fig 10 is very important since, minimal-ABC trees may contain two B₁-branches only in two very particular configurations.

Theorem 26. A minimal-ABC tree cannot contain two B₁-branches, unless the two B₁-branches belong to the configuration depicted in Fig 10 with s = 0, t = 1, or s = 0, t = 3.

proof. Suppose to the contrary that T is a minimal-ABC tree containing exactly two B₁-branches. Observe that the two B₁-branches are attached to the same vertex, say u, otherwise, there are at least two T_k-branches, which is a contradiction to Proposition 8. Moreover, by Proposition 15, u is not the root vertex of T. Denote by v the parent of u.

First, by Proposition 3, u contains no B_k-branch with k > 4. Next by Proposition 5, u contains no B₄-branch, and by Propositions 4 and 7, u contains no -branch, no matter u has B₃-branches or B₂-branches. Now we may deduce that the branches attached to u must be B₃-, B₂- or B₁-branches, i.e., T is of the structure depicted in Fig 10.

Set s and t for the numbers of B₃- and B₂-branches attached to u, respectively. Clearly, s + t ≥ 1, and s + t ≤ 10 from Proposition 23.

We apply the transformation depicted in Fig 15. And let d_x be the degree of vertex x in T.

Download:

Fig 15. The transformation

in the proof of Theorem 26.

https://doi.org/10.1371/journal.pone.0195153.g015

After applying , the degree of vertex v increases by s + t, while the degree of vertex u decreases by s + t, and the rest of the vertices do not change their degrees. The change of the ABC index after applying is (22)

Clearly, f(d_v + s + t, d_x) − f(d_v, d_x) < 0 for , and thus

On one hand, from Lemma 10, f(d_v + s + t, 3) − f(d_v, s + t + 3) increases in d_v, thus So it follows that (23)

Furthermore, note that d_v ≥ d_u = s + t + 3 from Proposition 2, and both f(d_v + s + t, 4) and f(d_v + s + t, 3) decrease in d_v ≥ s + t + 3, i.e., the right-hand side of (23) also decreases in d_v ≥ s + t + 3.

Besides the upper bound about ABC(T₁) − ABC(T) as (23), by considering a bit precisely in (22) for the term we may get a somewhat stricter upper bound about ABC(T₁) − ABC(T). Note that, from Lemma 10, f(d_v + s + t, d_x) − f(d_v, d_x) decreases in d_x, and from Proposition 2, every neighbor of v in has degree at least three, thus Now together with (22), it follows that (24)

Case 1. t = 0.

In this case, note that 1 ≤ s ≤ 10, and d_v ≥ s + 3 from Proposition 2.

By direct calculation, we may deduce that the right-hand side of (23) is negative, equivalently ABC(T₁) < ABC(T), holds for the following cases:

s = 1 and d_v ≥ 13;
s = 2 and d_v ≥ 17;
s = 3 and d_v ≥ 21;
s = 4 and d_v ≥ 25;
s = 5 and d_v ≥ 30;
s = 6 and d_v ≥ 35;
s = 7 and d_v ≥ 41;
s = 8 and d_v ≥ 47;
s = 9 and d_v ≥ 54;
s = 10 and d_v ≥ 61.

For the remaining cases as follows:

s = 1 and 4 ≤ d_v ≤ 12;
s = 2 and 5 ≤ d_v ≤ 16;
s = 3 and 6 ≤ d_v ≤ 20;
s = 4 and 7 ≤ d_v ≤ 24;
s = 5 and 8 ≤ d_v ≤ 29;
s = 6 and 9 ≤ d_v ≤ 34;
s = 7 and 10 ≤ d_v ≤ 40;
s = 8 and 11 ≤ d_v ≤ 46;
s = 9 and 12 ≤ d_v ≤ 53;
s = 10 and 13 ≤ d_v ≤ 60,

we would turn to use (24), and negative upper bounds, equivalently ABC(T₁) < ABC(T), follow from direct calculation easily.

Case 2. t = 1.

In this case, note that 0 ≤ s ≤ 9, and d_v ≥ s + 4 from Proposition 2.

By direct calculation, we may deduce that the right-hand side of (23) is negative, equivalently ABC(T₁) < ABC(T), holds for the following cases:

s = 1 and d_v ≥ 46;
s = 2 and d_v ≥ 38;
s = 3 and d_v ≥ 39;
s = 4 and d_v ≥ 43;
s = 5 and d_v ≥ 49;
s = 6 and d_v ≥ 55;
s = 7 and d_v ≥ 61;
s = 8 and d_v ≥ 68;
s = 9 and d_v ≥ 76.

For the remaining cases as follows:

s = 1 and 5 ≤ d_v ≤ 45;
s = 2 and 6 ≤ d_v ≤ 37;
s = 3 and 7 ≤ d_v ≤ 38;
s = 4 and 8 ≤ d_v ≤ 42;
s = 5 and 9 ≤ d_v ≤ 48;
s = 6 and 10 ≤ d_v ≤ 54;
s = 7 and 11 ≤ d_v ≤ 60;
s = 8 and 12 ≤ d_v ≤ 67;
s = 9 and 13 ≤ d_v ≤ 75,

we would turn to use (24), and negative upper bounds, equivalently ABC(T₁) < ABC(T), follow from direct calculation easily.

Case 3. t = 2.

In this case, note that 0 ≤ s ≤ 8, and d_v ≥ s + 5 from Proposition 2.

On one hand, the contradiction for s = 0 follows from Proposition 25.

On the other hand, by direct calculation, we may deduce that the right-hand side of (23) is negative, equivalently ABC(T₁) < ABC(T), holds for the following cases:

s = 1 and d_v ≥ 1402;
s = 2 and d_v ≥ 107;
s = 3 and d_v ≥ 84;
s = 4 and d_v ≥ 81;
s = 5 and d_v ≥ 84;
s = 6 and d_v ≥ 89;
s = 7 and d_v ≥ 96;
s = 8 and d_v ≥ 104.

For the remaining cases as follows:

s = 1 and 6 ≤ d_v ≤ 1401;
s = 2 and 7 ≤ d_v ≤ 106;
s = 3 and 8 ≤ d_v ≤ 83;
s = 4 and 9 ≤ d_v ≤ 80;
s = 5 and 10 ≤ d_v ≤ 83;
s = 6 and 11 ≤ d_v ≤ 88;
s = 7 and 12 ≤ d_v ≤ 95;
s = 8 and 13 ≤ d_v ≤ 103,

we would turn to use (24), and negative upper bounds, equivalently ABC(T₁) < ABC(T), follow from direct calculation easily.

Case 4. t = 3.

In this case, note that 0 ≤ s ≤ 7, and d_v ≥ s + 6 from Proposition 2.

On one hand, the contradiction for s = 1, 2 follows from Proposition 25.

On the other hand, by direct calculation, we may deduce that the right-hand side of (23) is negative, equivalently ABC(T₁) < ABC(T), holds for the following cases:

s = 3 and d_v ≥ 290;
s = 4 and d_v ≥ 193;
s = 5 and d_v ≥ 170;
s = 6 and d_v ≥ 163;
s = 7 and d_v ≥ 165.

For the remaining cases as follows:

s = 3 and 9 ≤ d_v ≤ 289;
s = 4 and 10 ≤ d_v ≤ 192;
s = 5 and 11 ≤ d_v ≤ 169;
s = 6 and 12 ≤ d_v ≤ 162;
s = 7 and 13 ≤ d_v ≤ 164,

we would turn to use (24), and negative upper bounds, equivalently ABC(T₁) < ABC(T), follow from direct calculation easily.

Case 5. t = 4.

In this case, note that 0 ≤ s ≤ 6. The contradiction may be deduced from Proposition 25.

Case 6. t = 5.

In this case, note that 0 ≤ s ≤ 5. The contradiction for the cases that s = 0 and s = 1, 2, 3, 4, 5 may be deduced from Propositions 24 and 25, respectively.

Case 7. t = 6.

In this case, note that 0 ≤ s ≤ 4. The contradiction for the cases that s = 0 and s = 1, 2, 3, 4 may be deduced from Propositions 24 and 25, respectively.

Case 8. t = 7.

In this case, note that 0 ≤ s ≤ 3. The contradiction for the cases that s = 0 and s = 1, 2, 3 may be deduced from Propositions 24 and 25, respectively.

Case 9. t = 8.

In this case, note that s = 0, 1, 2. The contradiction for the cases that s = 0 and s = 1, 2 may be deduced from Propositions 24 and 25, respectively.

Case 10. t = 9.

In this case, note that s = 0, 1. The contradiction for the cases that s = 0 and s = 1 may be deduced from Propositions 24 and 25, respectively.

Case 11. t = 10.

In this case, note that s = 0. The contradiction may be deduced from Proposition 24 directly.

Combining the above arguments, the result finally follows.

Discussion

The characterization of minimal-ABC trees is a rather active topic in chemical graph theory these years, which has led to a lot of structural properties and potential conjectures.

It is known that every pendent vertex of minimal-ABC trees belongs to some B_k-branch. As a strengthening, this paper proves that a minimal-ABC tree contains at most two B₁-branches. Moreover, we claim that a minimal-ABC tree can not contain two B₁-branches simultaneously, except for two cases that s = 0, and t = 1 or 3.

During the investigation of this paper, we also considered the two unsolved cases. However, to the best of our knowledge, until now we only get a solution under some particular degree conditions. In future research, the key point is to construct a more perfect graph transformation involve in general cases, which lead to a desired solution finally.

Actually, our ultimate goal is to show that the minimal-ABC trees contain no B₁-branch, when the order of that tree is large sufficiently.

References

1. Randić M. Characterization of molecular branching, J. Am. Chem. Soc. 97 (1975) 6609–6615.
- View Article
- Google Scholar
2. Estrada E. Torres L, Rodríguez L, Gutman I, An atom-bond connectivity index: Modelling the enthalpy of formation of alkanes, Indian J. Chem. 37A (1998) 849–855.
- View Article
- Google Scholar
3. Estrada E. Atom-bond connectivity and the energetic of branched alkanes, Chem. Phys. Lett. 463 (2008) 422–425.
- View Article
- Google Scholar
4. Ke X. Atom-bond connectivity index of benzenoid systems and fluoranthene congeners, Poly. Arom. Comp. 32 (2012) 27–35.
- View Article
- Google Scholar
5. Yang J, Xia F, Cheng H. The atom-bond connectivity index of benzenoid systems and phenylenes, Inter. Math. Forum 6 (2011) 2001–2005.
- View Article
- Google Scholar
6. Ahmadi MB, Sadeghimehr M. Atom bond connectivity index of an infinite class NS₁[n] of dendrimer nanostars, Optoelectron. Adv. Mat. 4 (2010) 1040–1042.
- View Article
- Google Scholar
7. Furtula B, Graovac A, Vukičević D. Atom-bond connectivity index of trees, Discrete Appl. Math. 157 (2009) 2828–2835.
- View Article
- Google Scholar
8. Ahmadi MB, Hosseini SA, Zarrinderakht M. On large trees with minimal atom-bond connectivity index, MATCH Commun. Math. Comput. Chem. 69 (2013) 565–569.
- View Article
- Google Scholar
9. Gutman I, Furtula B. Trees with smallest atom-bond connectivity index, MATCH Commun. Math. Comput. Chem. 68 (2012) 131–136.
- View Article
- Google Scholar
10. Gutman I, Furtula B, Ahmadib MB, Hosseini SA, Nowbandegani PS, Zarrinderakht M. The ABC index conundrum, Filomat 27 (2013) 1075–1083.
- View Article
- Google Scholar
11. Lin W, Lin X, Gao T, Wu X. Proving a conjecture of Gutman concerning trees with minimal ABC index, MATCH Commun. Math. Comput. Chem. 69 (2013) 549–557.
- View Article
- Google Scholar
12. Liu J, Chen J. Further properties of trees with minimal atom-bond connectivity index, Abstr. Appl. Anal. 2014 (2014) 609208.
- View Article
- Google Scholar
13. Das KC, Mohammed MA, Gutman I, Atan KA, Comparison between atom-bond connectivity indices of graphs, MATCH Commun. Math. Comput. Chem. 76 (2016) 159–170.
- View Article
- Google Scholar
14. Dimitrov D. Extremal trees with respect to the atom-bond connectivity index, Bounds in Chemical Graph Theory, Mathematical Chemistry Monogrpahs No.20, K. C. Das, B. Furtula, I. Gutman, E. I. Milovanović, I. Ž. Milovanović (Eds.), Pages 53–67, 2017.
15. Dimitrov D. On structural properties of trees with minimal atom-bond connectivity index IV: Solving a conjecture about the pendent paths of length three, Appl. Math. Comput. 313 (2017) 418–430.
- View Article
- Google Scholar
16. Dimitrov D, Ikica B, Škrekovski R. Remarks on maximum atom-bond connectivity index with given graph parameters, Discrete Appl. Math. 222 (2017) 222–226.
- View Article
- Google Scholar
17. Gao Y, Shao Y. The smallest ABC index of trees with n pendent vertices, MATCH Commun. Math. Comput. Chem. 76 (2016) 141–158.
- View Article
- Google Scholar
18. Lin W, Chen J, Ma C, Zhang Y, Chen J, Zhang D, Jia F. On trees with minimal ABC index among trees with given number of leaves, MATCH Commun. Math. Comput. Chem. 76 (2016) 131–140.
- View Article
- Google Scholar
19. Gutman I, Furtula B, Ivanović M. Notes on trees with minimal atom-bond connectivity index, MATCH Commun. Math. Comput. Chem. 67 (2012) 467–482.
- View Article
- Google Scholar
20. Wang H. Extremal trees with given degree sequence for the Randić index, Discrete Math. 308 (2008) 3407–3411.
- View Article
- Google Scholar
21. Gan L, Liu B, You Z. The ABC index of trees with given degree sequence, MATCH Commun. Math. Comput. Chem. 68 (2012) 137–145.
- View Article
- Google Scholar
22. Xing R, Zhou B. Extremal trees with fixed degree sequence for atom-bond connectivity index, Filomat 26 (2012) 683–688.
- View Article
- Google Scholar
23. Dimitrov D. On structural properties of trees with minimal atom-bond connectivity index, Discrete Appl. Math. 172 (2014) 28–44.
- View Article
- Google Scholar
24. Du Z, Fonseca CM da, On a family of trees with minimal atom-bond connectivity index, Discrete Appl. Math. 202 (2016) 37–49.
- View Article
- Google Scholar
25. Dimitrov D, Du Z, Fonseca CM da, On structural properties of trees with minimal atom-bond connectivity index III: Trees with pendent paths of length three, Appl. Math. Comput. 282 (2016) 276–290.
- View Article
- Google Scholar
26. Dimitrov D, Du Z, Fonseca CM da, Some forbidden combinations of branches in minimal-ABC trees, Discrete Appl. Math. 236 (2018) 165–182.
- View Article
- Google Scholar
27. Dimitrov D. On structural properties of trees with minimal atom-bond connectivity index II—Bounds on B₁- and B₂-branches, Discrete Appl. Math. 204 (2016) 90–116.
- View Article
- Google Scholar
28. Hosseini SA, Ahmadi MB, Gutman I. Kragujevac trees with minimal atom-bond connectivity index, MATCH Commun. Math. Comput. Chem. 71 (2014) 5–20.
- View Article
- Google Scholar
29. Dimitrov D. Efficient computation of trees with minimal atom-bond connectivity index, Appl. Math. Comput. 224 (2013) 663–670.
- View Article
- Google Scholar
30. Lin W, Gao T, Chen Q, Lin X. On the minimal ABC index of connected graphs with given degree sequence, MATCH Commun. Math. Comput. Chem. 69 (2013) 571–578.
- View Article
- Google Scholar

[ref1] 1. Randić M. Characterization of molecular branching, J. Am. Chem. Soc. 97 (1975) 6609–6615.
View Article
Google Scholar

[2] View Article

[3] Google Scholar

[ref2] 2. Estrada E. Torres L, Rodríguez L, Gutman I, An atom-bond connectivity index: Modelling the enthalpy of formation of alkanes, Indian J. Chem. 37A (1998) 849–855.
View Article
Google Scholar

[5] View Article

[6] Google Scholar

[ref3] 3. Estrada E. Atom-bond connectivity and the energetic of branched alkanes, Chem. Phys. Lett. 463 (2008) 422–425.
View Article
Google Scholar

[8] View Article

[9] Google Scholar

[ref4] 4. Ke X. Atom-bond connectivity index of benzenoid systems and fluoranthene congeners, Poly. Arom. Comp. 32 (2012) 27–35.
View Article
Google Scholar

[11] View Article

[12] Google Scholar

[ref5] 5. Yang J, Xia F, Cheng H. The atom-bond connectivity index of benzenoid systems and phenylenes, Inter. Math. Forum 6 (2011) 2001–2005.
View Article
Google Scholar

[14] View Article

[15] Google Scholar

[ref6] 6. Ahmadi MB, Sadeghimehr M. Atom bond connectivity index of an infinite class NS₁[n] of dendrimer nanostars, Optoelectron. Adv. Mat. 4 (2010) 1040–1042.
View Article
Google Scholar

[17] View Article

[18] Google Scholar

[ref7] 7. Furtula B, Graovac A, Vukičević D. Atom-bond connectivity index of trees, Discrete Appl. Math. 157 (2009) 2828–2835.
View Article
Google Scholar

[20] View Article

[21] Google Scholar

[ref8] 8. Ahmadi MB, Hosseini SA, Zarrinderakht M. On large trees with minimal atom-bond connectivity index, MATCH Commun. Math. Comput. Chem. 69 (2013) 565–569.
View Article
Google Scholar

[23] View Article

[24] Google Scholar

[ref9] 9. Gutman I, Furtula B. Trees with smallest atom-bond connectivity index, MATCH Commun. Math. Comput. Chem. 68 (2012) 131–136.
View Article
Google Scholar

[26] View Article

[27] Google Scholar

[ref10] 10. Gutman I, Furtula B, Ahmadib MB, Hosseini SA, Nowbandegani PS, Zarrinderakht M. The ABC index conundrum, Filomat 27 (2013) 1075–1083.
View Article
Google Scholar

[29] View Article

[30] Google Scholar

[ref11] 11. Lin W, Lin X, Gao T, Wu X. Proving a conjecture of Gutman concerning trees with minimal ABC index, MATCH Commun. Math. Comput. Chem. 69 (2013) 549–557.
View Article
Google Scholar

[32] View Article

[33] Google Scholar

[ref12] 12. Liu J, Chen J. Further properties of trees with minimal atom-bond connectivity index, Abstr. Appl. Anal. 2014 (2014) 609208.
View Article
Google Scholar

[35] View Article

[36] Google Scholar

[ref13] 13. Das KC, Mohammed MA, Gutman I, Atan KA, Comparison between atom-bond connectivity indices of graphs, MATCH Commun. Math. Comput. Chem. 76 (2016) 159–170.
View Article
Google Scholar

[38] View Article

[39] Google Scholar

[ref14] 14. Dimitrov D. Extremal trees with respect to the atom-bond connectivity index, Bounds in Chemical Graph Theory, Mathematical Chemistry Monogrpahs No.20, K. C. Das, B. Furtula, I. Gutman, E. I. Milovanović, I. Ž. Milovanović (Eds.), Pages 53–67, 2017.

[ref15] 15. Dimitrov D. On structural properties of trees with minimal atom-bond connectivity index IV: Solving a conjecture about the pendent paths of length three, Appl. Math. Comput. 313 (2017) 418–430.
View Article
Google Scholar

[42] View Article

[43] Google Scholar

[ref16] 16. Dimitrov D, Ikica B, Škrekovski R. Remarks on maximum atom-bond connectivity index with given graph parameters, Discrete Appl. Math. 222 (2017) 222–226.
View Article
Google Scholar

[45] View Article

[46] Google Scholar

[ref17] 17. Gao Y, Shao Y. The smallest ABC index of trees with n pendent vertices, MATCH Commun. Math. Comput. Chem. 76 (2016) 141–158.
View Article
Google Scholar

[48] View Article

[49] Google Scholar

[ref18] 18. Lin W, Chen J, Ma C, Zhang Y, Chen J, Zhang D, Jia F. On trees with minimal ABC index among trees with given number of leaves, MATCH Commun. Math. Comput. Chem. 76 (2016) 131–140.
View Article
Google Scholar

[51] View Article

[52] Google Scholar

[ref19] 19. Gutman I, Furtula B, Ivanović M. Notes on trees with minimal atom-bond connectivity index, MATCH Commun. Math. Comput. Chem. 67 (2012) 467–482.
View Article
Google Scholar

[54] View Article

[55] Google Scholar

[ref20] 20. Wang H. Extremal trees with given degree sequence for the Randić index, Discrete Math. 308 (2008) 3407–3411.
View Article
Google Scholar

[57] View Article

[58] Google Scholar

[ref21] 21. Gan L, Liu B, You Z. The ABC index of trees with given degree sequence, MATCH Commun. Math. Comput. Chem. 68 (2012) 137–145.
View Article
Google Scholar

[60] View Article

[61] Google Scholar

[ref22] 22. Xing R, Zhou B. Extremal trees with fixed degree sequence for atom-bond connectivity index, Filomat 26 (2012) 683–688.
View Article
Google Scholar

[63] View Article

[64] Google Scholar

[ref23] 23. Dimitrov D. On structural properties of trees with minimal atom-bond connectivity index, Discrete Appl. Math. 172 (2014) 28–44.
View Article
Google Scholar

[66] View Article

[67] Google Scholar

[ref24] 24. Du Z, Fonseca CM da, On a family of trees with minimal atom-bond connectivity index, Discrete Appl. Math. 202 (2016) 37–49.
View Article
Google Scholar

[69] View Article

[70] Google Scholar

[ref25] 25. Dimitrov D, Du Z, Fonseca CM da, On structural properties of trees with minimal atom-bond connectivity index III: Trees with pendent paths of length three, Appl. Math. Comput. 282 (2016) 276–290.
View Article
Google Scholar

[72] View Article

[73] Google Scholar

[ref26] 26. Dimitrov D, Du Z, Fonseca CM da, Some forbidden combinations of branches in minimal-ABC trees, Discrete Appl. Math. 236 (2018) 165–182.
View Article
Google Scholar

[75] View Article

[76] Google Scholar

[ref27] 27. Dimitrov D. On structural properties of trees with minimal atom-bond connectivity index II—Bounds on B₁- and B₂-branches, Discrete Appl. Math. 204 (2016) 90–116.
View Article
Google Scholar

[78] View Article

[79] Google Scholar

[ref28] 28. Hosseini SA, Ahmadi MB, Gutman I. Kragujevac trees with minimal atom-bond connectivity index, MATCH Commun. Math. Comput. Chem. 71 (2014) 5–20.
View Article
Google Scholar

[81] View Article

[82] Google Scholar

[ref29] 29. Dimitrov D. Efficient computation of trees with minimal atom-bond connectivity index, Appl. Math. Comput. 224 (2013) 663–670.
View Article
Google Scholar

[84] View Article

[85] Google Scholar

[ref30] 30. Lin W, Gao T, Chen Q, Lin X. On the minimal ABC index of connected graphs with given degree sequence, MATCH Commun. Math. Comput. Chem. 69 (2013) 571–578.
View Article
Google Scholar

[87] View Article

[88] Google Scholar

Figures

Abstract

Introduction

Preliminaries and methods

Lemmas

The root of B1-branches

Switching transformation

Results

The existence of four B1-branches

The existence of three B1-branches

The existence of two B1-branches

Discussion

References

The root of B₁-branches

The existence of four B₁-branches

The existence of three B₁-branches

The existence of two B₁-branches