Abstract
Let be the forgotten topological index of a graph . The exponential of the forgotten topological index is defined as , where is the number of edges joining vertices of degree and . Let be the set of trees with vertices; then, in this paper, we will show that the path has the minimum value for over .
1. Introduction
In this paper, let and be the vertex set and edge set, respectively. Let be the degree of a vertex in graph . A vertex of degree one is a pendant vertex or a leaf. A branching vertex of a tree is a vertex of degree .
Let be the path graph with length of ; if and , then we called is a pendant path.
Recently, topological indices have been considered by many researchers due to their many applications in various sciences. The forgotten topological index is defined in [1] as follows:
For applications of the forgotten topological index, see [2–4].
Before starting a new definition, we consider the set , and let be the number of edges joining vertices of degree and in a graph . Therefore, the new definition will be as follows:
The exponential of the forgotten topological index , denoted by , is defined as
Recently, the exponential topological indices have attracted the attention of many researchers. In [5], the exponential Randić index is characterized. In [6], the authors have characterized the exponential atom bond connectivity and the exponential augmented Zagreb index. In [7], the problem maximal value of trees for the exponential second Zagreb index is solved. Then, in this paper, we solve the problem with finding the minimal value of among trees.
2. Trees with Minimum Exponential of the Forgotten Topological Index
In this section, we will show that the path has the minimal value of the exponential forgotten topological index among all trees.
Lemma 1. Let and be the trees in Figure 1 and be a subtree of . If , then .
Proof. By setting , hence, we can write
Lemma 2. Let be a tree with minimum value of in and let be a pendant vertex , . If , then .
Proof. Suppose and be the largest path of and contains . Let be an end vertex of and a vertex in , where ; hence, by applying Lemma 1, we have .
We continue the proof with the following two cases.
Case 1. If .
Assuming that be the tree in Figure 2 and , where are the degrees of the adjacent vertices to different from . Hence, we can writeThis contradicts the minimality of .
Case 2. If .
Suppose and , where , , and . By applying Lemma 1, we get and . Let be the tree described in Figure 3. Therefore, we can writeHere, we showSinceandthe above inequality holds for . Therefore, for , we have . Hence, we get . That is a contradiction; hence, we get .
Let be a branching vertex of degree of a tree ; hence, can be viewed as the coalescence of subtrees of at the vertex . We call are the branches of at (see Figure 4).
Definition 1. (see [5]). A branching vertex of tree is an outer branching vertex of if all branches of at except for possibly one are paths.
Lemma 3 (see [5]).. A tree has no outer branching vertex if and only if .
Lemma 4. Let be the trees in Figure 5 and be a subtree of , and . Then, .
Proof. By direct calculation, it is not difficult to see . Here, we let ; therefore, we have
Lemma 5. Let be the trees in Figure 6 and . Then, , where .
Proof. We describe the graph in Figure 7; let be the path of length . It is not difficult to see . Then, by repeated of Lemma 4, we get .
Corollary 1. Let be a tree with a unique outer branching vertex and every pendant path has length at least 2. Then, .
Proof. If has a unique outer branching vertex, hence, has the form trees in Figure 6. Let be the tree in Figure 8. If , then . If , then by using Lemma 5, we have . Hence, we can write
Lemma 6. Let be the trees in Figure 9, such that ; then, .
Proof. By setting , we can write
Lemma 7. Let be the tree in Figure 10, and ; then, is not minimal in .
Proof. Set , and let be tree in Figure 11. Hence, we have if the following conditions are hold:(1).(2). It is not difficult to see that our result holds for . Therefore, we let . Then, To continue the proof, we must consider the following conditions:(3) and .(4) and .(5) and .(6) and .(7) and .(8) and .Note that, in (3), (4), and (5),, we have ; therefore, by Lemma 6, we can obtain trees with the minimum value of .
Here, if (6) holds, then we consider graph in Figure 12. Hence, we can writeIf (7) holds, then we consider graph in Figure 13. So, we haveFinally, if (8) holds, then we consider graph in Figure 14. Hence, we can writeTherefore, is not minimal in .
Theorem 1. Let and ; then, is not minimal for .
Proof. By using Lemma 3, we know that has an outer branching vertex . Using Lemmas 2, 5, and 6, we let all pendant paths of have length at least 2 and has the form in Figure 15, such that , and otherwise, is not minimal. If is the unique outer branching vertex of , then the result obtained by Corollary 1. Otherwise, among all outer branching vertices of , choose as the farthest from . From Lemma 5, we let is the form in Figure 16, such that . Note that is the farthest outer branching vertex from ; it is clear if is not a path; then, is an outer branching vertex of , and by Lemma 5, we let have the form in Figure 17. Therefore, we get , where is described in Figure 18 and . The result follows from Lemma 7.
Data Availability
No data were used to support the findings of this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.