#### Abstract

In this work, by using the properties of the variable sum exdeg indices and analyzing the structure of the quasi-tree graphs and unicyclic graphs, the minimum and maximum variable sum exdeg indices of quasi-tree graphs and quasi-tree graphs with perfect matchings were presented; the minimum and maximum variable sum exdeg indices of unicyclic graphs with given pendant vertices and cycle length were determined.

#### 1. Introduction

Topological indices are mathematical descriptors reflecting some structural characteristics of organic molecules on molecular graphs, and they play an important role in pharmacology, chemistry, etc. ([1–3]). For a graph , the variable sum exdeg index (denoted by ) was proposed by Vukičević [4] and is defined aswhere and is the degree of vertex . This graph invariant has a good correlation with the octanol-water partition coefficient [4] and was used to study the octane isomers given by the International Academy of Mathematical Chemistry (IAMC) [5–7]. Yarahmadi and Ashrafi [8] proposed a polynomial form of this graph invariant which is applied in nanoscience. By using the technique of majorization, Ghalavand and Ashrafi [9] provided the maximal and minimal (for ) of trees, bicyclic graphs, unicyclic graphs, and tricyclic graphs.

All graphs considered in this work are simple connected graphs. Let be a graph with the vertex set and the edge set . We denote by the minimum degree of . We use to denote the neighbourhood of a vertex and to denote the number of vertices with degree . Denoted by and the graphs arisen from by deleting the edge and by adding the edge , respectively. We denote by the subgraph of resulted by deleting the vertex with its incident edges. We call a quasi-tree graph if there is a vertex in such that is a tree. A unicyclic graph is the graph with exactly one cycle. Let and be two vertex-disjoint graphs. We denote by the graph having vertex set and edge set . As usual, we use , , and to denote the -vertex path, the -vertex star, and the -vertex cycle, respectively. The readers should refer for other definitions to [10].

There are many papers on the mathematical properties of topological indices, such as [11–14], since these invariants can detect the desirable properties of chemical molecules. In this work, we studied the mathematical properties of . This article is structured as follows. In Section 2, we present some useful lemmas. In Section 3, we obtain the maximal and minimal (for ) of quasi-tree graphs. In Section 4, we determine the maximal and minimal (for ) of quasi-tree graphs with perfect matchings. In Section 5, we derive the maximal and minimal (for ) of unicyclic graphs with given cycle length. In Section 6, we find the maximal and minimal (for ) of unicyclic graphs with given pendant vertex.

#### 2. Preliminaries

Lemma 1 (see [6]). *Let , where . Then*(i)* is strictly monotone increasing in *(ii)* and is strictly convex*

By Lemma 1, we have Lemmas 2 and 3 immediately.

Lemma 2. *Suppose is a connected graph, then*(i)*If , for *(ii)*If , for *

Lemma 3. *Let be positive integers with and . Then for , we have*

By simple calculation, Lemma 4 is immediate.

Lemma 4. *Letwhere , . Then is strictly monotone increasing in .*

Lemma 5. *Letwhere and . Then .*

*Proof. *Note thatSo, . Let , where . ThenThus, . So, for .

#### 3. Variable Sum Exdeg Indices of Quasi-Tree Graphs

Suppose is a quasi-tree graph and is a vertex in such that is a tree. If , then is a tree with extremal variable sum exdeg index (for ), that had been presented in [6, 9]. Thus, we always consider the case of in this section. Let.

.

Let be the graph arisen from complete bipartite graph by adding one edge between the two nonadjacent vertices with degree , as shown in Figure 1. We can easily obtain that .

Lemma 6. *Suppose such that has the maximal value of for . Let such that is a tree. Then, and .*

*Proof. *If , then there exists such that . Clearly, . In view of Lemma 2, , a contradiction. Therefore , and it can be concluded that .

Theorem 1. *Let , where . Then, for ,with the left equality if and only if and with the right equality if and only if .*

*Proof. *By induction on . When , it follows that and (8) holds. Assume that and (8) holds for .

First, we obtain the lower bound. If there is no pendant vertex in , since , then there exists such that . Let . For , let . By (1) and induction hypothesis, for , we havewith equality holding only if . This implies .

For , let . By of Lemma 1 and induction hypothesis, for , we haveOtherwise, there is at least one pendant vertex in . Let and . Then, . We denote by the vertex with . It can be seen that . If , then . By (1) and induction hypothesis, for , we haveIf , then by (1), (2), Lemma 4, and induction hypothesis, for , it follows thatNext, we obtain the upper bound. Choose such that has the maximum for . By Lemma 6, . Then, there is a vertex in such that since is a quasi-tree graph. By Lemma 6, it follows that . Denote . If , then has no edges. This implies that . If one of the vertices , say , satisfies , then . By (1), Lemma 1, and induction hypothesis, we havewhere , , , and , .

In [6], Vukičević obtained the minimal and maximal of trees on vertices for . The result is shown below.

Theorem 2 (see [6]). *Suppose is a tree on vertices, then for ,where the left equality holds only when , and the right equality holds only when .*

Thus, by simple calculation, we can extend our result to the whole quasi-tree graphs, as follows.

Theorem 3. *Let be an -vertex quasi-tree graph. Then, for ,where the left equality holds if and only if and the right equality holds if and only if .*

#### 4. Variable Sum Exdeg Indices of Quasi-Tree Graphs with a Perfect Matching

Let be the tree of order arisen from by adding a pendant edge to its pendant vertices, as shown in Figure 2. Let be the tree of order arisen from by adding a pendant edge to its every pendant vertex, as shown in Figure 2. Let and .

**(a)**

**(b)**

Lemma 7. *Let be positive integers. Then, for ,*

*Proof. *By (1) (2) and Lemma 4, for , we havesince .

Theorem 4. *Let be a quasi-tree graph of order with a perfect matching, where . Then, for ,with equality only when .*

*Proof. *When , (as shown in Figure 3). By Lemma 2, we have , *i* = 1, 2, 3.

If , choose such that has the maximal value of for . Assume that is a perfect matching of . We can suppose that is a tree since are quasi-tree graphs. Choose such that .

**(a)**

**(b)**

**(c)**

**(d)**

*Claim 1. *For any vertex of , .

The proof is similar to Lemma 6 (thus omitted).

*Claim 2. *For any vertex of except , .

To the contrary, assume that there is such that . Let and , where . By Claim 1, and . Since is a tree, we suppose that is the unique path from to in . Assume without loss of generality that (maybe or ). Notice that . Without loss of generality, assume that . Let . Clearly, is also a quasi-tree graph of order with a perfect matching. By (1) and (2),a contradiction with the choice of the graph .

By Claim 2, is a tree with some pendant paths attached to .

*Claim 3. *.

On the contrary, assume that . By the choice of , , thus and is a path on vertices. Denote . By Claim 1, , . It is not difficult to get that . By Lemma 5, for and , we havea contradiction with the choice of the graph .

We denote by () the paths attached to in .

*Claim 4. * for in .

To the contrary, suppose without loss of generality that in . Denote , where and . Then, there is at least one edge satisfying and . Let . Obviously, is also a quasi-tree graph of order with a perfect matching. By (1) and (2),a contradiction with the choice of the graph .

Denote . Since has a perfect matching, by Claim 4, it follows that or .

If , then . If , then . By (16), for , . Therefore, .

By Theorem 3, Theorem 5 is obtained immediately.

Theorem 5. *Suppose is a quasi-tree graph of order with a perfect matching, where , then for ,with equality if and only if .*

#### 5. Variable Sum Exdeg Indices of Unicyclic Graphs with Given Cycle Length

Let and (as shown in Figure 4) denote the graph obtained from by identifying its one vertex with the center vertex of and the graph obtained from by identifying its one vertex with a pendant vertex of , respectively.

**(a)**

**(b)**

Theorem 6. *Let be an -vertex unicyclic graph with cycle length . Then, for ,with equality only when .*

*Proof. *Choose such that has the maximum for . Suppose is the only cycle in .

*Claim 1. *There is at most one vertex with in .

To the contrary, suppose that there exist two vertices such that . Thus, there exists one vertex , but . It is evident that . Let . Then, has no change and is also an -vertex unicyclic graph with cycle length . By (1) and (2), it follows thata contradiction with the choice of the graph .

*Claim 2. *For , .

Assume, to the contrary, that there exists one vertex with . Denoted by (where and ) the path from to . Then . Since and , it follows that and . Denote , where . Let . Then, has no change and is also an -vertex unicyclic graph with cycle length . By (1) and (2), it follows thata contradiction again.

By Claims 1 and 2, we have .

Theorem 7. *Let be a unicyclic graph of order with cycle length . Then, for ,**with equality only when .*

*Proof. *Choose such that has the minimum for . Suppose is the only cycle in .

*Claim 3. * contains at most one pendant vertex.

Suppose that contains at least two pendant vertices. Let be two pendant vertices. We denote by (where , and ) the path from to with minimum length. Then, there is , such that and . Obviously, .

Let . Since and , then . Thus has no change and is also an -vertex unicyclic graph with cycle length . By (1) and (2), it follows thata contradiction with the choice of the graph .

Since , by Claim 3, has exactly one pendant vertex. This implies .

#### 6. Variable Sum Exdeg Index of Unicyclic Graphs with Given Pendant Vertex

Let (as shown in Figure 5) be the graph obtained from by identifying its one vertex with the center vertex of .

Let be the -vertex unicyclic graphs having pendant vertices and degree sequence , where .

Theorem 8. *Let be an -vertex unicyclic graph with pendant vertices. Then, for ,the equality holds only when .*

*Proof. *Choose such that has the maximum for .

*Claim 1. *There is at most one vertex with in .

Assume that there exist two vertices with . Let (where , ) be the path from to with minimum length. Since , there exists a vertex . Let . Clearly, is also an -vertex unicyclic graph with pendant vertices. In view of (1) and (2), it follows thata contradiction with the choice of the graph .

Since , by Claim 1, we have .

Theorem 9. *Let be an -vertex unicyclic graph with pendant vertices. Then, for ,where , with equality if and only if .*

*Proof. *Choose such that has the minimum for . Suppose is the only cycle in .

*Claim 2. *If are two nonpendant vertices of , then .

Assume that there are two vertices with . Suppose without loss of generality that . Since , then there exist at least two vertices . Furthermore, since is a unicyclic graph, contains at most one of . Set and . Let . Note that and , so is also a unicyclic graph with pendant vertices. In view of (1) and (2), it follows thata contradiction with the choice of the graph .

By Claim 2, we can find that has degree 1, , or , where . HenceSince is a unicyclic graph, then andBy (32) and (33), we have . By (32), , hence .

We also can get that .

So, has the degree sequencewhere .

#### 7. Results and Discussion

As one of the 148 topological indices that turned out good predictive properties, has a good correlation with the octanol-water partition coefficient. The mathematical properties of are worth studying [6] since this invariant can detect the desirable properties of chemical molecules. Therefore, our results may be used to predict the extremal properties of organic molecules.

#### 8. Conclusions

In this work, we present the minimum and maximum () of quasi-tree graphs and quasi-tree graphs with perfect matchings and determine the minimum and maximum () of unicyclic graphs with given pendant vertices and cycle length. We will consider the bicyclic graphs with some graph parameters for further study.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This work was funded by the Shanxi Province Science Foundation for Youths (grant no. 201901D211227).