#### Abstract

For a graph , its variable sum exdeg index is defined as , where is a real number other than 1 and is the degree of a vertex . In this paper, we characterize all trees on vertices with first three maximum and first three minimum values of the index. Also, we determine all the trees of order with given diameter and having first three largest values of the index.

#### 1. Introduction

We start by defining some basic notions related to graph theory. All the graphs we consider in this article are finite, simple, and connected. Let be a graph with vertex set and edge set . A tree is a graph without any cycle. Let denote the degree of a vertex and be defined as the count of neighbors of . A vertex of degree one is called pendent vertex. A path is a defined on a sequence of vertices with every vertex in the sequence adjacent to the vertex next to it. In particular, (unless ). Let and be the star graphs on and vertices, respectively. Denote by the graph obtained by identifying one of the end vertices of with the central vertex of . Also, is a graph obtained by identifying the central vertex of with one of the end vertex of path . The degree sequence of a graph is denoted by , and if the degree sequence of is , then we write . Furthermore, means has vertices of degree for . For undefined terminologies and notations, we refer the reader to .

A topological index is a numerical quantity derived from the graph of a molecule. It is well known that these indices are invariant under graph isomorphism. Topological index of a molecule determines many of its physical/chemical properties such as molecular volumes, electronic population, and energy volumes, see [2, 3], for details. A large number of topological indices have been defined through the years, and they are very well correlated with many physical/chemical properties of molecules . For a molecular graph , the variable sum exdeg index was proposed by Vukicevic  in 2011 to predict the octanol-water partition coefficient of certain compounds. It is denoted by and is defined aswhere is a real number other than 1. From the above definition, it can be seen that . It follows from the definition that the value of the index of any two graphs with the same degree sequence is equal. Among the benchmark set of 102 descriptors proposed by the International Academy of Mathematical Chemistry, 148 discrete Adriatic indices, and 48 variable Adriatic indices, the descriptor has the greatest coefficient of determination (namely, 0.99) for predicting the octanol-water partition coefficient of octane isomers . Therefore, it is of interest to explore the mathematical properties of this index. For , Vukicevic  determined the graphs with extremal values of the index among the classes of connected graphs (chemical graphs), unicyclic graphs (chemical unicyclic graphs), and trees (chemical trees). He also characterizes the extremal graphs with minimum (maximum) degree and trees having fixed number of vertices of degree 1. The variable sum exdeg polynomial was introduced by Yarahmadi and Ashrafi , and the effect of this polynomial under some graph operations was studied. They also studied the behavior of some nanotubes and nanotori using the variable sum exdeg polynomial. Using majorization technique, Ghalavand and Ashrafi  computed the graphs with extremal index value among the class of all vertex unicyclic graphs and tree. They have also given a complete characterization of graphs with maximum (for ) value in the class of -vertex tricyclic and bicyclic graphs. An alternate proof of some of the results in  is given by Ali and Dimitrov . They also extended the results for tetracyclic graphs. Recently, S. Khalid and A. Ali  attempted to find the graphs with the extremal index value among the trees with prescribed vertex degrees. For more details on the mathematical properties of index, we refer the readers to .

In this paper, first, we give an alternate proof to determine the trees with extremal values of index. Using the same construction, we found the second and third maximum/minimum values of the index. Next, we characterize the chemical trees on vertices with , where , having first and second maximum values of . Finally, we found the trees with the first three largest index values in class of trees on vertices and diameter .

#### 2. Extremal Values for Varibale Sum Exdeg Index of Trees

Lemma 1. Let be the graph with degree sequence . Suppose there exists a pair with and let a new graph be obtained from by changing the pair with . If , then .

Proof. Using the Mean value theorem and the definition of the index, we havewhere and . If and , then , and from equation (2), we get .

Lemma 2. Let be the graph with degree sequence . Suppose there exists a pair such that , and let a new graph be obtained from by changing the pair with . If , then .

Proof. Using the mean value theorem and the definition of the index, we havewhere and . Since and , then , and from equation (3), we get

Theorem 1. Let and be a tree of order . Then,(1) attains its maximum value iff (2) attains its 2nd maximum value iff (3) attains its 3rd maximum value iff (4) attains its minimum value iff (5) attains its 2nd minimum value iff (6) attains its 3rd minimum value iff

Proof. (1)Suppose the degree sequence of is . If is not isomorphic to , then there exists a pair with . Let be a tree obtained from by changing the pair with , then it follows from Lemma 2 that . We repeat the above procedure unless no pair is left with . In this way, we get a sequence of trees such that . Clearly, with degree sequence , and for any tree not isomorphic to , .(2)Above construction shows that is obtained from by changing the pair with , where . Observe that and .(3)Similarly, the degree sequence of has two cases: and . Note that can be obtained from by replacing the pair of by . Clearly, and . By Lemma 1, we have . Hence, .(4)Let be a tree of order with degree sequence . Suppose is not isomorphic to ; then, there exists a pair with . Let be a tree obtained from by changing the pair with the pair . Then, by Lemma 1, . Repeat the same operation until there is no pair such that for all . In this way, we get a sequence of trees such that and . Hence, for any tree not isomorphic to , .(5)Note that the degree sequence of is . Now, is obtained from by changing the pair with the pair , where . It is clear that the degree sequence of is and .(6)Similarly, has the following cases: or . Note that can be obtained from by replacing the pair by the pair . Then, by Lemma 1, . Hence, .From the above theorem, we get the following corollary.

Corollary 1. (see ). Let be a tree of order . Then,

Theorem 2. Let be a chemical tree of order and , where . If , then we have(1) attains its maximum value iff (2) attains its 2nd maximum value iff for for , and for

Proof. (1)Let be a chemical tree of order and let be its degree sequence. Suppose there exists a pair such that . We construct a graph from by changing the pair with the pair . Then, by Lemma 2, we have . We repeat the above procedure unless no pair is left with . In this way, we get a sequence of trees such that . Note that has exactly one vertex of degree 2 or degree 3, while the remaining vertices are of degree 4 or degree 1. Let be the vertices of degree 1, degree 2, degree 3, and degree 4, respectively; then,From the above equations, we get the following solutions:(1), , and if (2), , and if (3), , and if The solutions shows that attains its maximum value if and only if .(2)If , then from the proof of (2), it follows that . One can see that is obtained by by changing the pair with the pair , where . So, we have . If , then has the following two cases: or . Note that can be obtained from by replacing the pair by the pair . Then, by Lemma 1, . Hence, . The case for follows in the same way.From the above theorem, we get the following corollary.

Corollary 2. Let be a chemical tree of order and , where . Then, we have

In the next theorem, we characterize all the trees with the first three largest index values in class of trees with diameter and order .

Theorem 3. Let and be a tree of order with and . Then,(1) attains its maximum value iff (2)For , attains its 2nd maximum value iff (3)For , attains its third maximum value iff and for , attains its 3rd maximum value iff

Proof. The case is trivial. Suppose that . Let be a path of length on and be a vertex of maximum degree on . If , then there exists a vertex satisfying(a) and (b), , and First, suppose that there exists a vertex and . Choose such that is adjacent to exactly pendent vertices. Let the neighbors of be , and let .

Claim. :.
If in , then the claim follows from Lemma 2. Suppose in and let be the tree obtained from by changing the pair with the pair by pendent vertices (of ) transformation. Note that in . Then, it is easy to see that . Now, the tree can be obtained from the tree by applying the successive transformations and changing the pair with the pair . Thus, by Lemma 2, if follows . Hence, for , we have . This proves our claim.
We repeat the above process until there is no vertex in satisfying (a). Finally, we obtain a tree which has no vertex satisfying (a) and .(1)Next, we suppose that there is a vertex satisfying (b) for (instead of ). Let the neighbors of be , where and lie on the path . Let for . By Lemma 2, . Repeating the above operation, we get a sequence of trees on vertices and diameter such that . Note that, in , there is no pair of distinct vertices with degree greater or equal to three. Clearly, . This proves (1).(2)Suppose . Since and is obtained from by changing the pair with the pair , where , it is easy to see that has two cases: and . Note that can be obtained from by replacing the pair by the pair . Then, by Lemma 1, . This proves (2).(3)In the similar way, we get for and for . Since can be obtained from by replacing the pair by the pair , by Lemma 1, . It follows that for and . This proves (3).A vertex is called central vertex if . Every tree has either one or two central vertices. If , then the tree has just one central vertex, and if , then has two central vertices. If we choose a tree with given radius and , then we get the following result.

Theorem 4. Let and be a tree of order with and . Then,(1) attains its maximum value iff (2)For , attains its 2nd maximum value iff (3)For , attains its 3rd maximum value iff and for , and attains its 3rd maximum value iff

#### 3. Conclusion

In this paper, we have considered the variable sum exdeg index and studied its mathematical properties. For , we have characterized a class of trees having order and first three maximum/minimum values of the index. Also, we determine all the trees of order with given diameter and having first three largest values of the index. However, for , it remains an open problem to find the trees which have the first three maximum/minimum values of the variable sum exdeg index.

#### Data Availability

No data were used to support the study.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.