Abstract
The hyper-Zagreb index is an important branch in the Zagreb indices family, which is defined as , where is the degree of the vertex in a graph . In this paper, the monotonicity of the hyper-Zagreb index under some graph transformations was studied. Using these nice mathematical properties, the extremal graphs among -vertex trees (acyclic), unicyclic, and bicyclic graphs are determined for hyper-Zagreb index. Furthermore, the sharp upper and lower bounds on the hyper-Zagreb index of these graphs are provided.
1. Introduction
A topological index of a graph is a number with this property that, for every graph isomorphic to , Top() = Top(). In 1947, Wiener determined the most widely known topological descriptor, the Wiener index [1]. He used it to determine physical properties of paraffin. The Wiener index of a graph is equal to the sum of distances between all pairs of vertices of related graphs. Numerous indices have been explored. The Zagreb indices are the most important topological indices, introduced by Gutman and Trinajstić more than thirty years ago [2]. For a graph , the first and second Zagreb indices, and , respectively, are defined as In 1972, Zagreb indices first appeared in the topological formulas for the total -energy of conjugated molecules [2]. For applications in QSPR/QSAR, latest results are referred to [3–8].
In 2004, Miličević et al. [9] reformulated Zagreb indices in terms of edge-degrees instead of vertex-degree as follows: where is the degree of the edge in , defined by with , and means that the edges and are adjacent. Some results related to and are given in [10–12].
In 2013, Shirdel et al. [13] introduced a new degree-based topological index named hyper-Zagreb index as Recently, the multiplicative versions of Zagreb indices are studied well in [14]. Motivated by these results [15–18], we explore the properties for the hyper-Zagreb index.
Let be a simple and connected graph with vertex set and edge set . For a vertex , denotes the set of all neighbors of in . In a graph , the number of independent cycles is called its cyclomatic number and is equal to . Recall that graphs with are referred to as trees, unicyclic graphs, and bicyclic graphs, respectively. , , and denote the star, path, and cycle on vertices, respectively. Let , and then let be a subgraph of by deleting vertex and adjacent edges. For , let be a subgraph of by deleting an edge . Let be a nontrivial graph and let be its vertex. If is obtained from by fusing a tree at , then we say that is a subtree of and is its root. The fusions of two vertices and in are denoted by . In order to exhibit our results, we introduce some graph-theoretical notations and terminology. For other undefined ones, see the book [19].
In this paper, we characterize the extremal properties of the hyper-Zagreb index. In Section 2, we present some graph transformations which increase or decrease . In Section 3, we determine the extremal acyclic, unicyclic, and bicyclic graphs with maximum and minimum hyper-Zagreb index.
2. Graph Transformations
In this section, we will introduce some graph transformations, which increase or decrease the hyper-Zagreb index. These transformations will help to prove our main results. The following one from to strictly decreases the hyper-Zagreb index of a graph.
Transformation 1. Let be a nontrivial connected graph and is a given vertex in . Let be a graph obtained from by attaching two paths: of length and of length . If , we say that is obtained from by Transformation 1, as shown in Figure 1.
Lemma 1. If is obtained from by Transformation 1 as shown in Figure 1, then
Proof. In applying Transformation 1, the degree of decreases and the degrees of all its neighbors remain unchanged. So,
Transformation 2. Let be an edge of connected graph with . Suppose that are all the neighbors of vertex , while are pendant vertices. If , we say that is obtained from by Transformation 2, as shown in Figure 2.
Transformation 2 from to strictly increases of a graph.
Lemma 2. If K is obtained from by Transformation 2 as shown in Figure 2, then
Proof. Clearly, and is not changed during Transformation 2. Hence,
Transformation 3. Let be nontrivial connected graph and . Let == be a nontrivial -length path of connecting vertices and . If , we say that is obtained from by Transformation 3, as shown in Figure 3.
Lemma 3. If is obtained from by Transformation 3 as shown in Figure 3, then
Proof. From Figure 3, let and , while (merge and to obtain ) with , where . If , then If , then
Transformation 4. Let be a nontrivial acyclic subgraph of with which is attached at in graph ; let and be two neighbors of different from those in ; also and . If , we say that is obtained from by Transformation 4, as shown in Figure 4.
Lemma 4. Let G and K be two graphs, as shown in Figure 4. Then
Proof. From Transformation 2, we know that . So, we only prove the following inequality:Therefore, the proof is complete.
Let be a nontrivial connected graph. Two vertices and are said to be equivalent if . Clearly, and their neighbors have the same degree sequence.
Transformation 5. Let be a nontrivial connected graph and and are equivalent vertices in such that . Let be the graph obtained by attaching and at the vertices and of , respectively, with . If is the graph obtained by deleting the pendant vertices at in and connecting them to of , respectively, as shown in Figure 5. We say that is obtained from by Transformation 5.
Lemma 5. If K is obtained from by Transformation 5 as shown in Figure 5, then
Proof. We have
3. Main Results
In this section, we characterized the extremal graph with respect to among acyclic, unicyclic, and bicyclic graphs. First, we will define some notations which will be used later. denotes the set of all connected bicyclic graphs with order . Now we define three special classes of graphs. Let be the graph obtained by connecting two cycles and with a path with . Let be the graph which contains only two cycles and having a common vertex with , and let be the graph obtained by fusing two triples of pendant vertices of three paths , , and to two vertices with , where without loss of generality. Let be a bicyclic graph containing one of the three graphs , , and as its subgraph; then we will call it a brace of . We set , , and to be the set of all bicyclic graphs which include , , and as their brace, respectively. Clearly, , , and are the partitioned subsets of .
If is an acyclic graph with order , then, by Lemmas 1 and 2, the following result holds.
Theorem 6. Let G be an acyclic graph with order n. Then where the lower bound is achieved if and only if and the upper bound is achieved if and only if .
Theorem 7. Let be a unicyclic graph with vertices. Thenwhere the lower bound is achieved if and only if and the upper bound is achieved if and only if .
Proof. Being a unicyclic graph contains a unique cycle . By Lemma 3, we can obtain the graph in which the length of the cycle is three and its is increased strictly and then, by using Lemma 5, we can get the uniquely maximum graph with respect to (Figure 6). By using Lemmas 1 and 4, we find that the minimum graph is .
Theorem 8. Let be a bicyclic graph with vertices. Then where the lower bound is achieved if and only if and the upper bound is achieved if and only if .
Proof. By simple calculation, one can obtain . So, we show that if , then .
Case 1 ( Contains as Its Brace). As contains as its brace, by using Lemmas 2 and 5, we can obtain a new bicyclic graph whose is more than that of (see Figure 6). Clearly, .
Case 2 ( Is Not the Brace of ). Though does not contain the subgraph , by Lemma 3, maybe there is a bicyclic graph having brace whose is greater than . So we have two subcases.
Subcase 2.1 ( Is the Brace of ). By Lemma 3, Subcase is converted to Case .
Subcase 2.2 ( Is Not the Brace of ). By Lemmas 2, 3, and 5, we get a new bicyclic graph (Figure 6) whose is more than that of . It is easy to verify that .
Now we obtain the lower bound of bicyclic graphs with respect to . From Lemmas 1, 2, and 4, we find that the extremal graph of the minimum in bicyclic graphs must be element from the set .
Clearly, , , and .
So, we check the lower bound and equality holds if and only if .
Competing Interests
The authors declare that there are no competing interests regarding the publication of this paper.
Acknowledgments
This work was supported in part by the State Grid Anhui Economic Research Institute (no. 1P12001500010681000000) and Natural Science Foundation of Anhui Province of China (nos. KJ2015A331 and KJ2013B105).