Sharp Bounds of the Hyper-Zagreb Index on Acyclic , Unicylic , and Bicyclic Graphs

1School of Information Science and Technology, Yunnan Normal University, Kunming 650500, China 2Riphah Institute of Computing and Applied Sciences (RICAS), Riphah International University, Lahore, Pakistan 3Abdus Salam School of Mathematical Sciences, Government College University, Lahore, Pakistan 4Department of Applied Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran 5Department of Mathematics and Computer Science, Adelphi University, Garden City, NY 11530, USA 6School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China


Introduction
A topological index Top() 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,  1 and  2 , 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][4][5][6][7][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 () = () + (V) − 2 with  = V, and  ∼  means that the edges  and  are adjacent.Some results related to  1 () and  2 () are given in [10][11][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][16][17][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 V ∈ ,   (V) denotes the set of all neighbors of V in .In a graph , the number of independent cycles is called its cyclomatic number and is equal to  =  − +1.Recall that graphs with  = 0, 1, 2 are referred to as trees, unicyclic graphs, and bicyclic graphs, respectively.  ,   , and   denote the star, path, and cycle on  vertices, respectively.Let V ∈ , and then let  − V be a subgraph of  by deleting vertex V and adjacent edges.For  ∈ , let  −  be a subgraph of  by deleting an edge .Let  be a nontrivial graph and let V be its vertex.If  is obtained from  by fusing a tree  at V, then we say that  is a subtree of  and  is its root.The fusions of two vertices  and V in  are denoted by  + V.In order to exhibit our results, we introduce some graphtheoretical 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.

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 V is a given vertex in .Let  be a graph obtained from  by attaching two paths: (4) Proof.In applying Transformation 1, the degree of V decreases and the degrees of all its neighbors remain unchanged.So, ( Transformation 2. Let V be an edge of connected graph  with   (V) ≥ 2. Suppose that {V, Proof.Clearly,   (V) <   (V) and (()+(V)) is not changed during Transformation 2. Hence, Transformation 3. Let  be nontrivial connected graph and . ., V  }, we say that  is obtained from  by Transformation 3, as shown in Figure 3.
Proof.From Transformation 2, we know that () ≥ ( 1 ).So, we only prove the following inequality: Therefore, the proof is complete.
Let  be a nontrivial connected graph.Two vertices  and V are said to be equivalent if  −  ≅  − V. Clearly, |()| = |(V)| and their neighbors have the same degree sequence.
Transformation 5. Let  0 be a nontrivial connected graph and  and V are equivalent vertices in  0 such that   0 () =   (V) = .Let  be the graph obtained by attaching  +1 and  +1 at the vertices  and V of  0 , respectively, with  ≥  ≥ 1.
If  is the graph obtained by deleting the  pendant vertices at V in  and connecting them to  of , respectively, as shown in Figure 5.We say that  is obtained from  by Transformation 5. 5, then  () <  () .

Lemma 5. If K is obtained from 𝐺 by Transformation 5 as shown in Figure
(13) Proof.We have

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  +  +  − 2 = .Let   (, ) be the graph which contains only two cycles   and   having a common vertex with  +  − 1 = , and let   (, , ) be the graph obtained by fusing two triples of pendant vertices of three paths   ,   , and   to two vertices with  +  +  − 4 = , where 2 ≤  ≤  ≤  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  1  ,  2  , and  3  to be the set of all bicyclic graphs which include  ,,  ,   (, ), and   (, , ) as their brace, respectively.Clearly,  1  ,  2  , and  3  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 16 − 30 ≤  () ≤  2 ( − 1) , (15) 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.Then where the lower bound is achieved if and only if  ≅   and the upper bound is achieved if and only if  ≅  1  .
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  1  with respect to  (Figure 6).By using Lemmas 1 and 4, we find that the minimum graph is   .6) whose  is more than that of .It is easy to verify that ( 2 ) <  3 − 5 2 + 16 + 4. 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, ( ,,  ) = 16+70, (  (, )) = 16+96, and (  (, , )) = 16n + 70.
we say that  is obtained from  by Transformation 1, as shown in Figure1.Lemma 1.If  is obtained from  by Transformation 1 as shown in Figure 1, then  () <  () .