Further Properties of Trees with Minimal Atom-Bond Connectivity Index

and Applied Analysis 3


Introduction
Let  = (, ) be a finite, simple, and undirected graph.The degree of a vertex  ∈  is denoted by   .The atom-bond connectivity (ABC) index is defined as the sum of weights ((  +  V − 2)/   V ) 1/2 over all edges V of ; that is, The ABC index of a graph was defined by Estrada et al. [1] and it has many chemical applications [1,2].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 and what are these extremal values.In the case of the ABC index, finding the tree for which this index is maximal was relatively easy [3]; it is the star.Eventually, also the trees with second-maximal, third-maximal, and so forth ABC index were determined [4].
We [5] have shown that by deleting an edge from any graph, the ABC index decreases.This result implies that among all -vertex graphs, the complete graph   has maximal ABC value.Further, among all connected -vertex graphs, minimal ABC is achieved by some tree.Thus the vertex trees with minimal ABC index are also the -vertex connected graphs with minimal ABC index.But the problem of characterizing the -vertex trees with minimal ABC index turned out to be much more difficult, and a complete solution of this problem is not known.For more results on ABC index see [6][7][8][9][10][11][12][13].
In a recent work [6] a combination of computer search and mathematical analysis was undertaken, aimed at elucidating the structure of the minimal ABC trees.And some structural features of the trees with minimal ABC index are given in [7].
Lemma 1 (see [6]).If  ≥ 10, then the -vertex tree with minimal ABC index contains at most one pendent path of length  = 3. Lemma 2 (see [7]).If  ≥ 10, then each pendent vertex of the -vertex tree  with minimal ABC index belongs to a pendent path of length , 2 ≤  ≤ 3.
By inspecting the structural features of these trees, in [8] the branches  1 , . . .,  5 and  * 3 were given.Let   be a branch of tree  formed by attaching  pendant path of length 2 to the vertex V such that the degree of V in  is  + 1.Let  *  be a branch of tree  formed by attaching  − 1 pendant path of length 2 and a pendant path of length 3 to the vertex V such that the degree of V in  is  + 1 (see Figure 1).Denote by   the  union of the branches   and by (  ) the number of branches   in .From Lemmas 1 and 2, we know all branches in a tree  with minimal ABC index must be of the type   or  *  , and ( *  ) ≤ 1,  = 1, 2, . ... According to Lemma 1, in the following we assume that ( * 3 ) ≤ 1 and ( *  ) = 0, for all  ̸ = 3.
In [9] the -vertex minimal ABC trees were determined up to  = 300 and then a conjecture about the trees with minimal ABC index was presented.
In this paper, we determined a few structural features of the trees with minimal ABC index, also we characterized the trees with dia[] = 2 and minimal ABC index, where dia[] is the diameter of [], which was induced by the vertices of degree greater than 2 in .

The Structural Features of the Trees with Minimal ABC Index
Now, we are going to determine a few structural features of the trees with minimal ABC index.
Theorem 4. The -vertex tree with minimal ABC index does not contain branches   and  *  ( ≥ 6).
Proof.Suppose that  1  is a tree with minimal ABC index, possessing a branch   ,  ≥ 6.Let  be a vertex of  1  , adjacent to the vertex V, and the degree of  is .Consider the tree  2  (see Figure 3).By direct calculation, we have . ( If  = 6, it can be easily checked by computer that For the case  ≥ 7, if the inequality ABC( 1  ) > ABC( 2  ) holds, it implies that By elementary calculation, this inequality can be transformed to That is, By squaring the above relation and rearranging, we get Abstract and Applied Analysis Since the function and it holds that Figure 4 thus, we have In the same way, we can prove that the -vertex tree with minimal ABC index does not contain branch  *  ( ≥ 6).The proof is complete.
Note that Theorem 4 holds for all -vertex trees with minimal ABC index.Theorem 5. Let  be a tree with minimal ABC index, then every vertex of  must not be connected with both  2 and  4 .
Proof.Suppose that  1 is a tree with minimal ABC index; let  be a vertex of  1 , which is connected with both  2 and  4 , and the degree of  is  ( ≥ 3).Construct the tree  2 by deleting the edge V 2  and connecting  with V 1 (see Figure 4).
The transformation  1 →  2 causes the following change of the ABC index: If the inequality ABC( 1 ) > ABC( 2 ) holds, it implies that By elementary calculation, this inequality can be transformed to Thus we have ABC( 1 ) > ABC( 2 ), for  ≥ 3.
The proof is complete.
Theorem 6.Let  be a tree with minimal ABC index; then every vertex of  must not be connected with both  1 and 2 4 .
Proof.Suppose that  3 is a tree with minimal ABC index; let  be a vertex of  3 , which is connected with both  1 and 2 4 , and the degree of  is  (obviously  ≥ 3).Construct the tree  4 by deleting the edges V 2  2 , V 3  3 and adding the edges 5).The transformation  3 →  4 causes the following change of the ABC index: If the inequality ABC( 3 ) > ABC( 4 ) holds, it implies that That is ( Thus we have ABC( 3 ) > ABC( 4 ), for  ≥ 3, and the proof is complete.

Theorem 7.
Let  be a tree with minimal ABC index; then every vertex of  must not be connected with 7 4 .Proof.Suppose that  7 is a tree with minimal ABC index, possessing a vertex  in  7 connected with 7 4 (see Figure 6).Let  = {V 1 , V 2 , . . ., V  } be the set of adjacent vertices to .Let  1 ,  2 , . . .,   be the degree of V 1 , V 2 , . . ., V  , respectively.We consider the tree  8 shown in Figure 6.
Thus we have ABC( 7 ) > ABC( 8 ), and the proof is complete.
Theorem 8. Let  be a tree with minimal ABC index; then every vertex of  must not be connected with 2 5 .
Proof.Suppose that  1  5 is a tree with minimal ABC index, possessing a vertex connected with 2 5 .Let  be the vertex of  1  5 , adjacent to the vertices V 1 and V 2 , and the degree of  is  ( ≥ 3).Consider the tree  2 5 (see Figure 7).The transformation  1 5 →  2 5 causes the following change of the ABC index: It can be easily checked by computer that ABC( 1 5 ) > ABC( 2  5 ), for  ≥ 3. The proof is complete.

The Minimal ABC Indices of Trees with
Order  and dia[] = 2 Denote by [] the subgraph of  induced by its vertices of degree greater than 2. For a connected graph , the diameter of , denoted by dia, is the length of a longest path of .
The transformation  3 5 →  4  5 causes the following change of the ABC index: It can be easily checked that ABC( 3 5 ) > ABC( 4  5 ), for   ≥ 3. The proof is complete.Lemma 11.Let  ∈  ,2 be a tree with minimal ABC index; if the maximum degree Δ ≥ 24, then  must not contain  2 ( ≥ 3).
The transformation  5 →  6 causes the following change of the ABC index: By Theorems 4 and 5, Lemma 10, and noticing that  ∈  ,2 , we know that   = 2, 3 or 4.And consider Putting this in the above expression, we get ABC ( If the inequality ABC( 5 ) > ABC( 6 ) holds, it implies that By elementary calculation, this inequality can be transformed to By squaring the above relation and rearranging for two times, we get The largest root of the above polynomial is 23.1742; therefore, the value of the above polynomial is positive for  > 23.1742.Thus we have ABC( 5 ) > ABC( 6 ) for  ≥ 24, and the proof is complete.Lemma 12. Let  ∈  ,2 be a tree with minimal ABC index; if the maximum degree Δ ≥ 13, then  must not contain  1 .

Figure 2 :
Figure 2: Types of trees with minimal ABC index correspond to Conjecture 3.