General Randi T , Sum-Connectivity , Hyper-Zagreb and Harmonic Indices , and Harmonic Polynomial of Molecular Graphs

1Department of Applied Mathematics, Iran University of Science and Technology (IUST) Narmak, Tehran 16844, Iran 2School of Information Science and Technology, Yunnan Normal University, Kunming 650500, China 3Department of Mathematics, Maharani’s Science College for Women, Mysore 570005, India 4Department of Mathematics, The National Institute of Engineering, Mysuru 570008, India 5School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China


Introduction
In this paper, we consider only simple connected graphs without loops and multiple edges.A connected graph is a graph such that there is a path between all pairs of vertices.Let  = (, ) be an arbitrary simple connected graph; we denote the vertex set and the edge set of  by () and (), respectively.For two vertices  and V of (), the distance between  and V is denoted by (, V) and defined as the length of any shortest path connecting  and V in .For a vertex V of (), the degree of V is denoted by  V and is the number of vertices of  adjacent to V.
In chemical graph theory, we have many invariant polynomials and topological indices for a molecular graph.A topological index is a numerical value for correlation of chemical structure with various physical properties, chemical reactivity, or biological activity [1][2][3].
One of the oldest topological indices or molecular descriptors is the Zagreb index that has been introduced more than forty years ago by Gutman and Trinajstić in 1972 [4].Now, we know that, for a molecular graph  = (, ), the first Zagreb index  1 () and the second Zagreb index  2 () are defined as Recently, a new version of Zagreb indices named Hyper-Zagreb index was introduced by Shirdel et al. in 2013 [5] and it is defined as We encourage the reader to consult  for historical background and mathematical properties of the Zagreb indices.

Advances in Physical Chemistry
In 1975, Randić proposed a structural descriptor called the branching index [31] that later became the well-known Randić molecular connectivity index.Motivated by the definition of Randić connectivity index based on the end-vertex degrees of edges in a graph defined as the sum of the weights (   V ) −1/2 of all edges V of , Later, the Randić connectivity index had been extended as the general Randić connectivity index, which is defined as the sum of the weights (   V )  (∀ ∈ Q) and is equal to Also, a closely related variant of Randić connectivity index called the sum-connectivity index was introduced by Zhou and Trinajstić in 2008 [32,33].The sum-connectivity index () is defined as The general sum-connectivity index of a graph  is equal to (∀ ∈ Q) In 1987 [34], Fajtlowicz introduced the Harmonic index () of a graph  which is defined as the sum of the weights 2(   V ) −1 of ∀V ∈  and is equal to The Harmonic index is one of the most important indices in chemical and mathematical fields.It is a variant of the Randić index which is the most successful molecular descriptor in structure-property and structure activity relationships studies.The Harmonic index gives somewhat better correlations with physical and chemical properties compared with the well-known Randić index.Estimating bounds for () is of great interest, and many results have been obtained.For example, Favaron et al. [35] considered the relationship between the Harmonic index and the eigenvalues of graphs, and Zhong [36][37][38] determined the minimum and maximum values of the Harmonic index for simple connected graphs, trees, unicyclic graphs, and bicyclic graphs and characterized the corresponding extremal graphs, respectively.It turns out that trees with maximum and minimum Harmonic index are the path   and the star   , respectively.
In this paper, we present explicit formula for the general Randić connectivity, general sum-connectivity, Hyper-Zagreb and Harmonic Indices, and Harmonic polynomial of some hydrocarbon molecular graphs.

Results and Discussion
In this section, we compute the general Randić connectivity, general sum-connectivity indices, the Hyper-Zagreb and Harmonic Indices, and Harmonic polynomial of a family of hydrocarbon molecules, which are called Polycyclic Aromatic Hydrocarbons PAH  (∀ ∈ N).
The Polycyclic Aromatic Hydrocarbons PAH  is a family of hydrocarbon molecules, such that its structure is consisting of cycles with length six (benzene).The Polycyclic Aromatic Hydrocarbons can be thought of as small pieces of graphene sheets with the free valences of the dangling bonds saturated by .Vice versa, a graphene sheet can be interpreted as an infinite PAH molecule.Successful utilization of PAH molecules in modeling graphite surfaces has been reported earlier [44][45][46][47][48][49][50][51][52] and references therein.Some first members and a general representation of this hydrocarbon molecular family are shown in Figures 1 and 2.
Theorem 1 (see [45]).Consider the Polycyclic Aromatic Hydrocarbons PAH  (∀ ∈ N).Then, the first and second Zagreb indices of PAH  are equal to
(ii) the general sum-connectivity index of PAH  is equal to Theorem 5. Consider the Polycyclic Aromatic Hydrocarbons PAH  .Then, (i) the Harmonic index of PAH  is equal to ∀ ∈ N: (ii) the Harmonic polynomial of PAH  is equal to ∀ ∈ N: Before presenting the main results, consider the following definition.
Definition 6 (see [10]).Let  be a simple connected molecular graph.We divide the vertex set () and edge set () of  based on the degrees  V of a vertex/atom V in .Obviously, 1 ≤  V ≤  − 1 and we denote the minimum and maximum of the  V by  and Δ, respectively: Proof of Theorem 2. Let PAH  be the Polycyclic Aromatic Hydrocarbon for all integer numbers .From the general representation of PAH  in Figure 2, one can see that in this hydrocarbon molecular family there are 6 2 + 6 vertices/atoms (= |(PAH  )|) such that 6 2 of them are carbon atoms and also 6 of them are hydrogen atoms.In other words, Thus, there are Now, by using Definition 6 and according to Figure 2, one can see that in hydrocarbon molecules PAH  all hydrogen atoms have one connection and the degree of them is  Hydrogen = 1 and there are 3 edges/chemical bonds for all carbon atoms; thus,  Carbon = 3.
Therefore, we have two partitions of the vertex set (PAH  ) of Polycyclic Aromatic.

Advances in Physical Chemistry
Hydrocarbons PAH  are as follows: Here, we complete the proof of Theorem 2.
Proof of Theorem 4. Consider the Polycyclic Aromatic Hydrocarbons PAH  with 6 2 +6 vertices/atoms and 9 2 +3 edges.Then, by using the results from the above proof, we have the following computations for the general Randić and sum-connectivity indices of PAH  (∀ ∈ N, ∀ ∈ Q): Here, the proof of Theorem 5 was completed.

Figure 1 :
Figure 1: Some first members of the Polycyclic Aromatic Hydrocarbons (PAH  ).